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Imposing classical symmetries on quantum operators with applications to optimization
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Imposing classical symmetries on quantum operators with applications to optimization
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Imposing Classical Symmetries on Quantum Operators with Applications to Optimization by Hannes Leipold A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (COMPUTER SCIENCE) May 2023 Copyright 2023 Hannes Leipold Dedicated to Lutz and Eunyoung. ii Acknowledgements I would like to thank my advisors Federico Spedalieri and Greg ver Steeg and Todd Brun and Aiichiro Nakano as part of the guidance committee. I want to give special thanks to Federico for many years of close mentorship and advising, and treating me like a colleague and friend rather than a student during all our research endeavors. Nothing in this thesis would be possible without his tireless support. I would also like to give special thanks to Eleanor Rieffel for her mentorship and support and Stuart Hadfield for helpful discussions. I thank all my close friends who have made USC a home for me during my studies and my family for their continuous encouragement. I also thank all the sources of funding that have allowedthisresearchtocometofruitionandlistourfundingsourcesintheacknowledgements of the individual chapters. iii Table of Contents Dedication ..................................................................................... ii Acknowledgements ............................................................................ iii List of Tables .................................................................................. vii List of Figures ................................................................................. viii Abstract........................................................................................ xii Chapter1: Introduction: ImposingClassicalConstraintsEnablesMoreTailoredQuan- tum Algorithms............................................................................ 1 1.1 Motivation............................................................................ 1 1.2 Constrained Quantum Annealing................................................... 2 1.2.1 Transverse Field Quantum Annealing Overview .......................... 3 1.2.2 Constrained Quantum Annealing Overview ............................... 4 1.3 Quantum Alternating Operator Ansatz............................................ 5 1.3.1 Quantum Approximate Optimization Algorithm Overview .............. 7 1.3.2 Quantum Alternating Operator Ansatz Overview ........................ 7 1.4 Scope of the Thesis.................................................................. 8 Chapter2: ConstructingDriverHamiltoniansandMixingOperatorsforOptimization Problems with Linear Constraints........................................................ 10 2.1 Introduction.......................................................................... 10 2.2 Background .......................................................................... 12 2.2.1 Constraint Quantum Annealing for Integer Linear Programming........ 13 2.3 An Algebraic Condition for Commuting with Linear Constraints................ 16 2.4 Bounded weight drivers ............................................................. 19 2.5 The Problem ILP-QCOMMUTE ........................................................ 21 2.5.1 Bounded Weight ILP-QCOMMUTE....................................... 23 2.6 Reachability within the Feasible Space............................................. 24 2.7 Conclusion............................................................................ 31 2.8 Acknowledgements for Chapter..................................................... 33 Chapter3: ConstructingDriverHamiltoniansandMixingOperatorsforOptimization Problems with General Constraints ...................................................... 35 iv 3.1 Introduction.......................................................................... 35 3.2 An Algebraic Condition for Commuting with Constraints........................ 36 3.3 Sufficient Condition for Quadratic Constraints.................................... 39 3.4 Imposing Constraint Symmetries on Quantum General and Bounded Weight Operators............................................................................. 41 3.5 Conclusion............................................................................ 42 3.6 Acknowledgements for Chapter..................................................... 43 Chapter 4: Quantum Annealing with Special Drivers for Circuit Fault Diagnostics..... 44 4.1 Introduction.......................................................................... 44 4.2 Combinatorial Optimization for Circuit Fault Diagnostics ....................... 46 4.2.1 The set of valid diagnoses................................................... 48 4.3 Constrained Quantum Annealing Approach....................................... 50 4.3.1 Special Drivers for CQA Solvers tackling CFD............................ 51 4.3.2 Implementing CQA protocols for CFD with special drivers.............. 53 4.4 Benchmarks on Synthetic Instances: Spectral Gaps and Simulated Annealing Schedules............................................................................. 55 4.4.1 Generalized ISCAS C17 Random Circuits: Spectral Gaps and Param- eterized Annealing Schedule ................................................ 55 4.4.1.1 Spectral Gaps for C17 Instances................................. 55 4.4.1.2 Choosing annealing schedule to exploit spectrum features..... 58 4.4.2 Generalized C26 Random Circuits: Effects of Degeneracy and Multi- Fault Solution Spaces ....................................................... 60 4.5 Conclusions........................................................................... 62 4.6 Acknowledgements for Chapter..................................................... 67 Chapter 5: Tailored Quantum Alternating Operator Ans¨ atze for Circuit Fault Diag- nostics ...................................................................................... 68 5.1 Introduction.......................................................................... 68 5.2 Constrained Evolution in Quantum Alternating Operator Ans¨ atze.............. 71 5.3 Circuit Fault Diagnostics............................................................ 74 5.3.1 Valid and Invalid Configurations Around Gates........................... 76 5.4 QAOA approaches to CFD.......................................................... 77 5.4.1 Approach 1: Transverse Field with QUBO................................ 78 5.4.2 Approach 2: Transverse and XY-mixer with QUBO...................... 79 5.4.3 Approach 3: Graph Diffusors with Linear Field on Fault Bits........... 81 5.4.4 Approach 4: Transverse Field on Faults with Oracle Circuit Simulator. 83 5.4.5 Approach 5: Bounded Fault Count with Oracle Circuit Simulator ...... 85 5.4.6 Size of the Relevant Constrained Space and Summarization for Ap- proaches...................................................................... 86 5.5 Random CFD instances with Balanced Width and Depth ....................... 88 5.6 Performance with Different Parameter Optimization ............................ 91 5.7 Conclusion............................................................................ 94 5.8 Acknowledgements for Chapter..................................................... 96 v Chapter 6: Conclusion: Satisfying Classical Symmetries Can Guide Future Quantum Algorithm Design.......................................................................... 97 6.1 Summary............................................................................. 97 6.2 Future Work ......................................................................... 99 References......................................................................................101 Appendices.....................................................................................109 C 0-1-LP-QCOMMUTE is NP HARD................................................110 C.1 Generalized Full Adder......................................................112 C.2 The Simple Reduced Case ..................................................116 C.3 The Simple Unreduced Case................................................119 C.4 The General Unreduced Case...............................................120 C.5 Proof of Runtime............................................................125 C.6 Reducing a Solution of ILP-COMMUTE to a Solution of EQUAL SUBSET SUM............................................................................125 D Proof of the Matrix Implementation of the Generalized Adder ..................127 vi List of Tables 4.1 Consistent assignments for a NAND gate: note that there is no restriction on thevaluesoftheinputs,theirfaultbitsandtheoutput(hence,wehave2 5 =32 configurations). However, the output fault bit is completely determined by the values of the inputs and the output. .......................................... 50 5.1 Each kind of input/output pair to a faulty NAND is described. Each entry has different valid configurations and configurations in the same entry form a subspace such that the action of the driver terms we construct connect the configurations of this subspace...................................................... 78 5.2 Depending on our selection of mixing and phase-separating operators, we can constraintheevolutionofthewavefunctiontoaconstrainedspacesofdiffering sizes. ................................................................................. 86 6.1 The generalized full adder; if u ∗ takes a particular value on inputs a and b, then u ∗ will be forced to take the corresponding sum (represented by s) and carry (represented by c) values. In the case that a+b is not a power of two, s and c have two possible values they can take. Here primary (secondary) op- erations correspond to the operations where the carry is set to zero (nonzero) if possible. ...........................................................................114 vii List of Figures 4.1 AschematicoftheC17circuitwithafaultonoutputofthethirdNANDgate for a given input/output pair. In Sec. 4.4, we generate new random circuits by replacing each NAND gate with a randomly selected two input logic gate... 47 4.2 (1) shows the instantaneous minimum gap for 100 random instances of the generalized C17 circuit at regular discretized points of time. (2) similarly shows the instantaneous ground state gap for the same instances with a non- stoquastic version of the driver Hamiltonian....................................... 56 4.3 (1) and (2) show the success probability versus the minimum gap for 22 non- degenerate single fault instances of the C17, with 40 units and 80 units of timerespectively,forthesingleparameterfunction(labeledparam),thelinear function(labeledlinear)andapiece-wiselinearfunctionfittedover100evenly spacedpoints, suchthattheslopeisproportionaltotheinstantaneousinverse gap squared between evenly spaced s i− 1 and s i (labeled opt adia)............... 59 4.4 A schematic of the C26 circuit. For our C26 benchmarks, we replace each NANDgatewitharandomlyselectedtwoinputlogicgateaswellasrandomly select inputs to the circuit. Each has 8 2-input logic gates, 6 FAN gates, 26 wires, 6 inputs, and 4 outputs....................................................... 60 viii 4.5 (1) shows a box and whisker plot of the logarithm of the minimum gap as a function of the number of faults in the MFD, while (2) shows a box and whisker plot of the location of the minimum gap versus the MFD number of faults, for randomly generated instances of C26 with non-degenerate ground states. ................................................................................ 61 4.6 Logarithm of success probability as a function of instance degeneracy and number of faults in the MFD (for different values of the total annealing time T f ). .................................................................................. 63 5.1 All possible valid fault configurations for a small circuit with one FAN gate and one NAND gate that has a faulty output. The diagrams on the top left and right show minimum fault explanations for this instance. ................... 75 5.2 A visual representation of the subspaces that the wavefunction is kept within during evolution through each QAOA approach considered in this chapter. .... 87 5.3 Diagrams for circuit instances of the CFD problem considered in Sec. 5.6, dependingonthenumberofqubits,fromthesesmall,local,anddensecircuits. Everyboxisatwointput/oneoutputgate,everydotisaoneinput/twooutput gate, and every diamond is a one intput/one output gate. If a top wire has a ⊤ and a bottom wire has a⊥ at the same depth, they refer to the same wire (which has been wrapped around). ................................................ 89 5.4 Performance of ans¨ atze with different parameter selections. The solid lines indicate the performance on the median of the instances, while the ribbons correspond to the lower and upper quantiles. Markers correspond to the success probabilities of individual instances........................................ 91 6.1 A Venn diagram of different complexity classes discussed in this thesis under the assumption that P̸=NP. ........................................................ 98 ix 6.2 A flow cart describing how our reduction works, we recommend motivated readers refer back to it as they read the reduction. An instance of EQUAL SUBSET SUM (box 1) is mapped into a binary constraint representation such thatthesumfunctionEdefinedovertheassignment uisequivalenttosum(A)− sum(B) where u assigns variables to either A, B, or they are not used (box 2). To exploit Theorem 2.3.1, constraints C are mapped to constraints K (box 3), such that assignment E K (u ∗ )=0⇔E S (u ∗ )=0. Unlike S,K allows for a simple matrix representation such that K M ⃗ µ ∗ = 0⇔ E K (u ∗ ) = 0 (box 4), where ⃗ µ is a naive vectorized form of u ∗ . Note that if u exists such that E S (u)=0, then many u ∗ exist such that E K (u ∗ )=0, but each reduces to the same u. The constraint version of K M can be embedded row-wise to define operators ˆ K 1 ,..., ˆ K S∪A as ˆ K i = P |S∪A| j=1 k M ij σ z j such that a 0-1-LP-QCOMMUTE oracle solves to show the existence of a driver Hamiltonian H d , which we can interpret back to see there must be a solution to EQUAL SUBSET SUM as well. ..113 6.3 SubfigureAshowsthereducedembeddingofthe EQUAL SUBSET SUM instance with the (improper) integer set {1,1,2}. Each box represents a generalized fulladder. Eachadderdescribesacorrespondingsubmatrixinthematrix ˜ K M (check Eq. C.8). Subfigure B shows the full embedding of the same instance...117 x 6.4 This figure shows the layout of the generalized complete adder for enforcing that u ∗ is only valid if the corresponding u on S ={s 1 ,...,s n } is also valid. In each row, a box corresponds to a generalized adder (with the truth table given in Table 6.1) where the output of that whole row (labeled by the sum bit z) is zero if and only if u ∗ is valid. After m (the largest bit length of any s i ∈ S) rows, the next rows are fed only carries from the previous rows. As such, the number of generalized adders decreases by one, until the very last row,wherewehavethatz mn 1 andk mn 1 shouldbothbezeroforu ∗ tobevalidon the set. The final constraint matrix is a representation of this diagram, with eachgeneralizedadderrepresentingasubmatrixthatenforcestherelationship shown in Table 6.1 (check Eq. C.1). ...............................................121 xi Abstract Applyingquantumcomputerstosolvecombinatorialoptimizationtasksisoneofthemost excitingwaystoleveragequantumsystemsforpracticalcomputationaladvantage. Heuristic quantum approaches to solve such problems have become important barometers to track the progress in developing Noisy Intermediate-scale Quantum (NISQ) processors, since they may not require as advanced fault-tolerance as Grover’s search or Shor’s algorithm and can be more amendable to hybridization techniques. Two of the primary paradigms to lever- age quantum computers for such tasks are Quantum Annealing (QA) and the Quantum AlternatingOperatorAnsatz(QAOA).Atypicalapproachforbothwouldbetomapacom- binatorial optimization problem with feasibility constraints to an unconstrained quadratic optimizationproblem. However, itispossibletoimposetheseconstraintsontheevolutionof the quantum system by selecting the quantum operators applied to the system to maintain the evolution such that the corresponding observables remain invariant under the evolution of the quantum state. We consider such approaches to be imposing classical symmetries on the quantum operators. In this thesis, we give an algebraic formulation to the problem of imposing an arbi- trary collection of constraint symmetries on quantum operators. This allows us to describe a general algorithm to solve the corresponding task for linear constraints and classify the complexity of several related computational problems. In particular, we show that finding a nondiagonalHamiltonianthatmaintainsacollectionofclassicalsymmetriesisNP-Hard,but xii thatthecorrespondingproblemforlinearconstraintsistractableforboundedweightHamil- tonians, which is particularly relevant since most quantum implementation are restricted to low-body interactions where utilizing such an algorithm is attractive. We then consider a quantum annealing protocol for the problem of combinational cir- cuit fault diagnostics (CCFD) and analyze features of our approach that make it attractive for quantum annealers built to solve this class of combinatorial optimization problems. In- deed, CFD, the optimization problem underlying CCFD, is a natural optimization problem to study for quantum annealers due to its natural topologically local lattice-like description. We consider several QAOA protocols that are tailored to impose different constraint sym- metries underlying this problem and study the trade-offs between the protocols. Our results are consistent with the view that tailoring the ansatz of a protocol to match the underlying symmetry of an optimization problem can be beneficial to finding solutions with a lower QAOA depth under several parameter optimization schemes. We hope the theoretical results and concrete constructions in this thesis can aid re- searchers in developing more domain specialized quantum algorithms for optimization tasks or search problems, although there may be broader applications of our key results. xiii Chapter 1 Introduction: Imposing Classical Constraints Enables More Tailored Quantum Algorithms 1.1 Motivation Over the last two decades, major strides in the development and control of quantum systems has brought quantum computation and quantum algorithms to the forefront of research development around the world. Many fundamental questions about the computa- tional resourcefulness of quantum systems in relation to classical systems remain open. Due to the intractability of combinatorial optimization problems [1–4], there is a great deal of interest in developing quantum approaches to try and tackle these type of problems. For the context of this thesis, we focus on two approaches for utilizing a quantum computer to solve combinatorial optimization problems: Quantum Annealing and Quantum Alternating OperatorAnsatzalgorithms. Thesetwoapproachesareverygeneralyetpowerfulparadigms to guide the development of quantum solvers for optimization tasks. These approaches are particularly attractive for noisy intermediate scale quantum (NISQ) devices [4–6]. For both, the four central features are (1) the initial state of the quantum system, (2) the family of operators to apply, (3) the control parameters, and (4) the search space utilized. We consider these features as part of our overview of both paradigms. As we will 1 see, the question of finding Hamiltonians with classical symmetries become a central design question for feature (2) and must inform the other features. 1.2 Constrained Quantum Annealing Quantum annealing (QA) [7–11] is a type of continuous time quantum evolution where a dynamic Hamiltonian evolves continuously. The most widely studied form considers inter- polations between different Hamiltonians. Typically, quantum annealing is used to undergo adiabaticquantumevolutionsuchthattheinstantaneouswavefunctionisclosetotheinstan- taneous ground state of the instantaneous Hamiltonian. For some finite annealing time T, we have a continuous smooth function to dimensionless time s(t) : [0,T]→ [0,1] such that the Hamiltonian at any time is given by: H(s(t))=A(s(t))H d +B(s(t))H f . (1.1) The instantaneous eigenvalues and eigenvectors of H(s) are E 1 (s) ≤ ... ≤ E D (s) and |E 1 (s)⟩,...,|E D (s)⟩ respectively such that H(s) = P i E i (s)|E i (s)⟩⟨E i (s)|. Then under slow (adiabatic) evolution, |ψ (s)⟩≈| E 1 (s)⟩ (see Refs. [12–15] for a discussion on various rigorous conditions). However, quantum annealing has also be explored for diabatic [16] evolution and can have varying degree of sensitivity to adiabaticity for finding solutions of specific problems (for example, for those with high ground space degeneracy). With transverse field quantum annealing (TFQA), the Hamiltonian at any moment can be described as an interpolation point between two Hamiltonians, H d = P n i=1 σ x i , and a final Hamiltonian H f according to Eq. 1.1. To impose that the solutions found to an optimization problem are feasible (i.e in the feasible subspace), typically a con- strainedproblemistransformedintoaquadraticunconstrainedbinaryoptimizationproblem (QUBO). In this case, the final Hamiltonian can be written as an Ising-spin Hamiltonian: H f = P n i=1 h i σ z i + P n i=1 P n j=i J ij σ z i σ z j . Indeed, quantum annealers designed specifically to 2 solve these type of problems have been studied and are some of the earliest quantum devices that are able to solve optimization problems, although they have limitations based on their connectivity, operator locality, decoherence time, and control of system parameters [17, 18]. Refs. [19, 20] introduced the notion of Constrained Quantum Annealing (CQA) as a newavenuetodoquantumannealingforproblemswithfeasibilityconstraints. Inparticular, the transverse field driver Hamiltonian is replaced with a different driver Hamiltonian such that, given a wavefunction inside the feasible subspace, the closed system dynamics will maintain a wavefunction throughout the evolution that has support only in the feasible subspace. ForseveralNP-Hardoptimizationproblems,thiswasshowntoreducethenumber of multi-body terms required between qubits and the geometrical locality of such terms as well as show a larger minimum gap during the anneal (thereby allowing a faster annealing under adiabatic assumptions). In this work the question was raised of finding a procedure or algorithmic approach that could be used to find general Hamiltonians that keep the feasibility space invariant. Chapter 2 answers this question for linear constraints while Chapter 3 focuses on general constraints. We give general overview descriptions of both TFQA and CQA which showcase the similarities and differences in the approaches to solving an optimization problem. 1.2.1 Transverse Field Quantum Annealing Overview 1. Initial State: The uniform superposition state |+⟩ N n = 1 √ 2 n P x∈{0,1} n |x⟩. With certain catalyst drivers, adaptive schemes, or reverse annealing this is replaced (usually to a classical state). 2. Driver Hamiltonian: The transverse field H d =− P n i=1 σ x i . 3. Cost Hamiltonian: An Ising-spin Hamiltonian from a QUBO description H f = P n i=1 h i σ z i + P n i=1 P n j=i J ij σ z i σ z j . 3 4. Annealing Protocol: Generally an interpolation between two Hamiltonians A(s(t))H d +B(s(t))H f . Generalized when involving catalyst drivers [21][22]. 5. Search Space: The entire Hilbert space C 2 n . 1.2.2 Constrained Quantum Annealing Overview 1. Initial State: Usually a classical state or a uniform superposition state over the fea- sible subspace (especially for a single uniformly weighted linear constraint). Note that the ground state of the driver Hamiltonian is not necessarily a uniform superposition state of the feasibile subspace, and different avenues have been considered for particular problems [19, 20, 23]. 2. Driver Hamiltonian: Hamiltonian H d such that [ ˆ C i ,H d ] = 0 for all constraints C i where ˆ C i is the corresponding constraint embedding operator for constraint C i . How can we find such a Hamiltonian? This is the central question answered in Chapters 2 and 3. A prototypical example is a constraint C = (1,...,1) such that the constraint embedding operator is ˆ C = P n i=1 σ z i and H XY = P n− 1 i=1 σ x i σ x i+1 +σ y i σ y i+1 commutes with ˆ C. 3. CostHamiltonian: HamiltonianH f associatedwiththeoptimizationfunctiontoopti- mizeoverthefeasiblesubspace. Sincethereisnoneedtoincludepenaltytermsassociated with enforcing feasibility, it involves fewer two-bodied terms in comparison to the cost Hamiltonian of TFQA. 4. Annealing Protocol: Generally an interpolation between two or more Hamiltonians A(s(t))H d +B(s(t))H f if the ground state of H d is easily preparable, but usually A(s(t))H in +B(s(t))H d +C(s(t))H f . Generalized when involving catalyst drivers. 5. Search Space: The feasible subspace span({|x⟩|x is a feasible solution}). 4 1.3 Quantum Alternating Operator Ansatz The Quantum Approximate Optimization Algorithm (QAOA1) was introduced in a landmark paper [24] and has, by any measure, taken the NISQ algorithms community by storm. A simple and flexible framework is lay out for using single qubit spin-x rotations Q n j=1 e − iβσ x j followed by a cost operator, usually using a phase-separating operator made of single qubit spin-z and two qubit spin-z operators: e iα ( P n j=1 h j σ z j + P j P k J jk σ z j σ z k ) = n Y j=1 e iαh j σ z j n Y j=1 n Y k=j e iαJ jk σ z j σ z k . (1.2) It is easy to see by their construction that these cost operators are rich enough, just as Ising spin models are for QA, to model combinatorial optimization problems with quadratic constraints. As such, an optimization problem, for example the Max Cut problem, can be translated into a QUBO that is then used as the phase-separating operator for QAOA1. Like in TFQA, the initial state considered in the original paper is the uniform superposition |+⟩ ⊗ n = 1 √ 2 n P x∈{0,1} n |x⟩. Given a discrete number of total rounds, p, for which to apply the phase-separating operatorandthenthemixingoperatoronthequantumsysteminasequenceoflength p, the taskistofindappropriateparameters α 1 ,...,α p andβ 1 ,...,β p tominimizetheexpectedcost oftheresultingwavefunctionwithrespecttothecostfunction. Thiscanbeformulatedasan optimizationprobleminitself. Forexample,onecanbeginwithaparticularsetofparameters (orrandomlysamplethem)andthenrunthequantumsystemusingtheseparameters. Based on the expected cost from the measurements, one can update the parameters as a form of localoptimization. AsQAOAandrelatedvariationalquantumalgorithms(VQA[6]),suchas variationalquantumeigensolvers(VQE),havebecomeafocalpointfortheNISQcommunity there have been many different methods considered for tackling this problem. Inparticular, ithasbeenshowntobeNP-Hardingeneral[6]tooptimizedtheparame- ters of a QAOA protocol and QAOA protocols can be plagued by a barren plateau problem, 5 which can lead to being stuck in nonoptimum local minimas. However, large scale simu- lation studies of small quantum systems show that for certain problems practically useful parameters for QAOA can follow simple patterns [25, 26] despite the lack of a guaranteed optimal selection (with adiabatic or diabatic features). Since VQAs for low depth circuits remain highly attractive for experimenting with NISQ devices, this remains an area of great consideration by the quantum research community. Quantum Alternating Operator Ansatz (QAOA) [27–29] explores the same consider- ations of CQA in the context of VQAs for the circuit model. Rather than utilizing the mixing operator of QAOA1, QAOA considers a generalization where the mixing operators and phase-separating operators are tailored to the feasible space by imposing classical sym- metries on the mixing operators (since the phase-separating operators are diagonal in the computationalbasis)suchthattheevolutionofthesystemwouldberestrictedtothefeasible subspace. As such, the same fundamental question arises in the context of finding unitary operators to commute with constraint embedding operators as Hamiltonians in CQA and the same results from Chapter 2 and Chapter 3 can be utilized in the context of finding appropriate mixing operators for QAOA. Ref. [29] includes a compendium of mappings for many NP-Hard problems with mixing operators satisfying classical symmetries associated with each optimization problem. We give an overview, similar to the overviews for TFQA and CQA, for QAOA1 and QAOA. Note the similarity between TFQA and QAOA1 as well as that of CQA and QAOA. However, we stress that QAOA is not necessarily analogous (via Trotterization) to Adia- batic Quantum Computation. For example, Grover’s search algorithm is a QAOA algorithm with Grover’s diffuser as the mixing operator and the marked state as the phase-separating operator. 6 1.3.1 Quantum Approximate Optimization Algorithm Overview 1. Initial State: The uniform superposition state |+⟩ N n = 1 √ 2 n P x∈{0,1} n |x⟩ in the original paper [24]. Other initial states have been studied in other settings [30]. 2. Mixing Operators: Spin-x associated unitaries U d (β ) = Q n j=1 e − iβσ x j . For finite p QAOA, each round has an angle β that is optimized. 3. Phase-separating Operators: Spin-z one body and two body terms U c (α ) = Q n j=1 e iαh j σ z j Q n j=1 Q n k=j e iαJ jk σ z j σ z k . For finite p QAOA, each round has an angle α that is optimized. 4. Round Unitary: U(α i ,β i )=U d (β i )U c (α i ) for round i in [1,...,p]. 5. Search Space: The entire Hilbert space C 2 n . 1.3.2 Quantum Alternating Operator Ansatz Overview 1. InitialState: Usuallyaclassicalstateorauniformsuperpositionstateoverthefeasible subspace (especially for a single uniformly weighted linear constraint [31]). 2. Mixing Operators: U d (β ) such that [U d (β ), ˆ C i ] = 0 for constraints C i where ˆ C i is the constraintembeddingoperator(seeChapter2). Howcanwefindsuchmixingoperators? This is the central question answered in Chapters 2 and 3. A prototypical example is a constraint C = (1,...,1) with U XY (β ) = Q n− 1 i=1 e − iβ (σ x i σ x i+1 +σ y i σ y i+1 ) (typically split into two non-commuting operators). 3. Phase-separating Operators: Phase-separating operators U c (α ) associated with the optimization function to optimize over the feasible subspace. Since no penalty terms associated with enforcing feasibility, it involves fewer two-bodied terms in comparison to the phase-separating operators of QAOA1. 7 4. Round Depth: U(α i ,β i )=U d (β i )U c (α i ) if no catalyst, otherwise U(α i ,β i ,γ i )=U in (γ i )U d (β i )U c (α i ). 5. Search Space: The feasible subspace span({|x⟩|x is a feasible solution}). 1.4 Scope of the Thesis In this thesis, fundamental results related to the task of imposing classical symmetries on quantum operators are rigorously proven in Chapter 2 and Chapter 3. We show that, for a general collection of constraints, the decision problem associated with imposing the associ- ated symmetries on a nontrivial operator is NP-Complete. This is even true for constraints with binary coefficients (a full proof is given in Appendix C). For an arbitrary collection of linear constraints (of polynomial cardinality) there is a straight forward polynomial time algorithm for local operators. For quadratic conditions we find a simple sufficient condition. This task plays an important role in designing driver Hamiltonians in the context of CQA or mixing operators in the context of QAOA such that they maintain classical symmetries and can be used in practice by researchers developing quantum algorithm for constrained optimization tasks. In Chapter 4, we consider the problem of combinational circuit fault diagnostics for CQA,whichhasbecomeaproblemofinterestfortheQAcommunity[4, 32, 33]duetobeing acombinatorialoptimizationproblemwithtypicalinstanceswithgrid-likestructuressimilar to practical quantum annealers. We construct a specialized driver Hamiltonian to maintain the constrained subspace of valid fault diagnostics and consider some of the features of this approach for solving CFD instances through simulation on two families of circuits that are small enough to benchmark through simulations. Our results are consistent with higher degeneracy and higher number of required faults in the solution space being associated with better performance of the CQA protocol for a fixed annealing time. 8 In Chapter 5, we consider different ans¨ atze for QAOA protocols, each utilizing a dif- ferent kind of symmetry in the problem. We consider the performance of each under a class of designed small circuits with different parameter selection methods. These results suggest developing tailored an¨ atze for problems with higher symmetry, such as CFD, can lead to practical advantages for low and medium range p. In Chapter 6, we summarize the results of this thesis and discuss exciting directions for future work. 9 Chapter 2 Constructing Driver Hamiltonians and Mixing Operators for Optimization Problems with Linear Constraints In this chapter, we fully introduce constrained quantum annealing (CQA) and give a general algebraic formula for reasoning about imposing classical symmetries on quantum operators, focusing primarily on its application for CQA. Our results translate immediately to the Quantum Alternating Operator Ansatz (QAOA) setting. We use this formulation to prove several important complexity classification related to imposing linear constraints on quantumoperatorsingeneralandcontextualizetheseresultsfortheapplicationinCQAand QAOA. This chapter is based on work in Ref. [34]. 2.1 Introduction Quantum annealing has been proposed as a heuristic method to exploit quantum me- chanical effects in order to solve discrete optimization problems. Typically, these problems require optimizing a quadratic cost function subject to a set of linear constraints. The usual approachtotreatingtheseconstraintsconsistsofaddingthemtothecostfunctionaspenalty terms, thereby transforming the constrained optimization into an unconstrained one. This approach has some drawbacks, including an increase in the required resources (i.e., higher 10 connectivity and increased dynamical range for the parameters that define the instance). In Ref. [19], the authors introduced the idea of constrained quantum annealing (CQA), that uses specially tailored driver Hamiltonians for a set of feasibility constraints. These tailored Hamiltonians have several advantages, such as reducing the size of the search space of the problem and reducing the number of needed interactions to implement the annealing proto- col. At the heart of the approach is the idea that a Hamiltonian which commutes with the operator embedding of the constraints and starts within the feasible space of configurations will remain in it throughout the evolution. While Ref. [19] looked primarily at commuting with a single global constraint, the work in Ref. [20] focused on finding driver Hamiltonians for several constraints. Under special conditions, the authors were able to construct appropriate driver Hamiltonians for several optimization problems of practical interest. In this chapter, we ask and answer the general question of, given a set of arbitrary linear constraints, can we construct a driver Hamiltonian which will commute with the operator embedding of the constraints? Our main result is that this problem is NP-Complete – answering a question posited originally in Ref. [20]. Along the way, we will derive a simple formula for describing the commutation relation and exploit it to understand many facets of CQA. These results can be naturally applied to the Quantum Alternating Operator Ansatz (QAOA) framework [27], where one of the central tasks is to find mixer operators that connect feasible solutions of a constrained optimization problem. Since unitary operators are exponentials of Hermitian operators(foreveryunitarymatrixU,thereexistsaHermitianmatrixH suchthatU =e iH ) and therefore the existence of a unitary matrix with this commutative property necessitates the existence of the Hermitian matrix with the same commutative property (since [e iH , ˆ C]= P ∞ k=0 [(iH) k , ˆ C]/k!), our results directly translate into this setting as well. The chapter is organized as follows. In Section 5.2 we review the basic ideas behind constrained quantum annealing. In Section 2.3, we derive a simple algebraic condition for the commutation relation of the driver Hamiltonian and the constraint operators. For many 11 practical applications, Hamiltonians with bounded weight interaction (local) terms are often desired; in Section 2.4 we show how brute forcing the simple algebraic condition from Sec- tion 2.3 to find driver Hamiltonians of bounded weight. In Section 2.5, we introduce several variations of the problem ILP-QCOMMUTE, the problem of finding a Hermitian matrix that willcommutewiththeconstraintoperators. Wealsoreducethe EQUAL SUBSET SUMproblem to the problem ILP-QCOMMUTE (and in Appendix C we show it for the special case of binary valued linear constraints). The EQUAL SUBSET SUM problem is known to be NP-Complete, thusprovingourmainresult. Wedefinearelatedcomplexityclass, ILP-QCOMMUTE-k-LOCAL, about finding a Hermitian matrix that commutes with the constraints and has interaction termsuptokweight. ILP-QCOMMUTE-k-LOCALisinPbytheapproachdetailedinSection2.4. Other questions of interest, such as finding a driver Hamiltonian that has full reachability over a constrained space for a CQA protocol, will be addressed through our formulation in Section 2.6. We conclude with a discussion of the significance of our result and open problems related to what we have shown in this chapter. 2.2 Background In the quantum annealing (QA) framework, gradually decreasing quantum fluctuations are used to traverse the barriers of an energy landscape in the search of global minima to complicated cost functions [8, 9]. For an overview of these approaches, we refer the reader to Ref. [12]. Quantum annealing has gained traction for combinatorial optimization [10, 11, 33, 35] as a way to solve hard optimization problems faster and, more recently, for machine learning[36–41]asawaytonaturallysampledesiredprobabilitydistributionsquickly. Inthe case of solving an optimization problem, the problem is encoded in the Hamiltonian H p such that the ground state is the optimum solution. Usually this is readily done by expressing the problem as an Ising model, a model for spin glasses [42–45]. 12 OncetheproblemHamiltonianisdescribed,theQAframeworkprescribesanevolution to the final Hamiltonian from some readily preparable Hamiltonian H d - usually through a linear interpolation of H d and H p : H(s)=sH p +(1− s)H d , (2.1) where there is a continuous smooth function s(t) for t ∈ [0,T] such that s(0) = 0 and s(T)=1. Iftheprocessisvariedslowlyenough,theadiabatictheoremensuresthatthewave functionofthesystemwillbeclosetotheinstantaneousgroundstateofthesystemforany s and therefore any t. By the adiabatic theorem, if the total time T that the system is evolved for is large compared to the inverse of the minimum gap squared, then the wavefunction of the system will be close to the ground state of H p . For the purpose of our presentation here, we restrict our focus to the case of binary linear optimization problems, a heavily studied optimizationclass. Specifically,wecanconsiderasetoflinearconstraints C ={C 1 ,...,C m }. Suppose that the solutions to the optimization problem are subject to constraints C i such that a solution state x∈{0,1} n satisfies C i (x) = P j c ij x i = b i for some b i . Because C i is a simple linear function, we can associate a vector ⃗ c i with it such that C i (x) =⃗ c i · ⃗ x where ⃗ c i ∈Z n . When referring to constraints throughout this chapter, we are referring specifically to linear constraints, for which our main results are pertinent to. 2.2.1 ConstraintQuantumAnnealingforIntegerLinearProgramming Weusetheordinaryembeddingofbinary variablesx i ∈{0,1}inthecomputationalbasis for quantum annealing, such that ⃗ x ∈ {0,1} n is represented by |⃗ x⟩ = |x 1 ⟩...|x n ⟩ ∈ C 2 n (i.e. σ z i |⃗ x⟩ = (1− 2x i )|⃗ x⟩) and the final Hamiltonian is diagonal in the computational basis so that we can read off a solution by measuring in that basis. Following the framework of CQA, given a constraint C(x) = ⃗ c· ⃗ x, we associate C with an embedded constraint operator ˆ C = P n i=1 c i σ z i . Let us consider the case of a single constraint - C = (1,...,1) - 13 over n variables. This is also the first case presented in Ref. [19]. It is simple to check that H d = P n− 1 i=1 (σ x i σ x i+1 +σ y i σ y i+1 )commuteswiththeconstraintembeddedoperator ˆ C = P n i=1 σ z i . Forexample,thistypeofconstraintmayariseingraphpartitioning,sincethepartitionsmust split the graph into equal size. For the Graph Partition problem, one is given a graph G and is asked to partition the vertices V into equal subsets such that the number of edges between the two is minimized. In terms of the Ising model, we can consider a collection of n qubits, such that|0⟩ (|1⟩) for qubit i represents placing vertex v i in partition 1 (2). As such, we design a penalty Hamiltonian H p and a driver Hamiltonian H d such that the final state will be a solution to the graph partitioning problem. Assuming the transverse field driver Hamiltonian - H d = P n i=1 σ x i - a simple penalty Hamiltonian can be: H p = X (i,j)∈E 1− σ z i σ z j +α n X i=1 σ z i ! 2 , (2.2) where the first term assigns a positive potentiality to each edge that connects vertices across the partitions and the second term is the constraint operator squared. In general the penalty factor α must be greater than min(2d m ,n)/8 where d m is the maximal degree of G [46]. Note that the term ( P n i=1 σ z i ) 2 is ˆ C 2 and requires n 2 two body interaction terms to implement. However, if we choose our H d such that [H d , ˆ C] = 0, then we can use the simpler penalty Hamiltonian: H p = X (i,j)∈E 1− σ z i σ z j . (2.3) Note that since H p is diagonal in the spin-z basis, it trivially commutes with the con- straints. One benefit to this construction is that the driver H d = P n− 1 i=1 (σ x i σ x i+1 +σ y i σ y i+1 ), for example,commuteswith ˆ C andonlyrequiresn− 1twobodyinteractiontermstoimplement. As such, the total number of two body terms required to solve the problem can be greatly 14 reduced by using driver Hamiltonians beyond the transverse field if they commute with a set of constraints. As long as the initial wavefunction has an expected value of n/2 for ˆ C (n/2 + 1 or n/2− 1 if n is odd), the wavefunction will remain in the subspace with this expected value for the entirety of the anneal. As an example of Graph Partition that we will return to later, consider a graph with 4 vertices V = {v 1 ,v 2 ,v 3 ,v 4 }, connected into a single path by edges E ={e 1 ,e 2 ,e 3 } with e 1 = (v 1 ,v 2 ),e 2 = (v 2 ,v 3 ),e 3 = (v 3 ,v 4 ). Then in this case, H d = (σ x 1 σ x 2 +σ y 1 σ y 2 )+(σ x 2 σ x 3 +σ y 2 σ y 3 )+(σ x 3 σ x 4 +σ y 3 σ y 4 ). Since we are interested in an even partition, the starting state should have an equal number of 1s and 0s. For example, |0011⟩ is in the correct subspace. It is important to note that driver Hamiltonians are constructed irrespective of the value that the constraint is set to, since the eigenvalue of the constraint operator with respect to the initial wavefunction will determine whatvalueispreservedduringtheanneal. Forawavefunctionthatisnotaneigenvectorofa constraint operator, the commuting property of the driver Hamiltonian with the constraint operator will mean that the projection of the wavefunction onto each specific eigenspace will evolve independently of the rest of the wavefunction. Forthepurposeofadiabaticquantumcomputing,theinitialwavefunctionofthesystem has to be in the ground state of the initial Hamiltonian, while a general driver Hamiltonian from this construction can be highly nontrivial if there are many constraints. Therefore, specifying an alternative initial Hamiltonian for CQA is also a major area of research, since wewanttheinitialHamiltoniantohaveagroundstatethatiseasytoprepare. Oneapproach to overcome this is to use an initial Hamiltonian diagonal in the computational basis and linearontheσ z operators,thathasasitsgroundstateaspecific solutioninthefeasiblespace in the spin-z basis, and then evolve from this initial Hamiltonian to the driver Hamiltonian (whose ground state will have support on all or a subset of the feasible space). There are many hard problems for which finding a feasible solution is simple. For example, it is straightforwardtofindasinglepartitionfor Graph Partition, butitishardtofindthe best partition. Assuch, ausefulavenueforexploitingCQAisinthecasewherelinearconstraints 15 specify a nontrivial feasibility space, but one where finding a nonoptimum element in the feasibility space is still tractable. TheworkinRef.[20]extendedtheframeworkforcaseswhereadrivershouldcommute with multiple constraint operators. In particular, given a set C, they consider finding a Hamiltonian H d such that: [H d , ˆ C j ]=0, C j ∈C (2.4) As they note, in general, tailoring driver Hamiltonians for a set C can be difficult. In this chapter, we answer specifically the computational complexity of such a task by reducing an NP-Complete problem to ILP-QCOMMUTE. We also discuss the related task of finding H d such that it can reach every state in the solution space, but reaches no state outside the solutionspaceinSection2.6. Thisresultinsomewaysmayappearintuitive,sincedescribing the feasible space ofC is hard and knowing a Hamiltonian that would keep a wavefunction within this space - and only this space - should require some characterization of it. Simply knowing that a nontrivial H d exists at all for a NP-Complete feasibility problem allows one to recognize that the problem should have at least two solutions for some set of values that each constraint is set to, even if one does not have a token to prove it. 2.3 AnAlgebraicConditionforCommutingwithLinear Constraints Consider the problem to find Hamiltonian drivers that have, as their eigenvectors, sup- port over the possible values that satisfy the given linear constraints. In the most general sense, we consider constraints of the form: ˆ C = n X i=1 c i σ z i , ⃗ c∈Z n , (2.5) 16 withaconstraintvaluebthatcorrespondstooneoftheenergylevelsof ˆ C. Oftenproblemsof practical interest can be captured in the restricted case where⃗ c∈{0,1} n or⃗ c∈{− 1,0,1} n . Consider the linear transformation [M,σ z ] that maps any two by two matrix M to a new two by two matrix M ′ , by commuting the matrix with σ z . This transformation has two obviouseigenmatrices- 1andσ z -thatspanthekernelofthetransformation. Onecaneasily verifythatσ + (|0⟩⟨1|)andσ − (|1⟩⟨0|)arealsoeigenmatrices ofthistransformation,with eigenvalue 2 and − 2 respectively. Together these four eigenmatrices and their eigenvalues describe the spectrum decomposition of the transformation. We exploit this fact to find a simple algebraic formula for expressing the commutation of a general Hamiltonian with a linear constraint. It is easy to verify that for H over n qubits, if Tr 1,...i− 1,i+1,...n [H] = σ ± , then [H,σ z i ]=± 2H. Givenanycompletebasisforasinglequbitsystem,wecanextendthatbasistodefinea basisovernqubits. Doingthiswiththefoundeigenmatricesdefinesabasis {1,σ z ,σ + ,σ − } ⊗ n . Noteaswellthat α j σ ± i † =α † j σ ∓ i forα j ∈C. Thissuggestsasimplerepresentationinwhich a Hermitian matrix is defined by its nonzero terms over this basis. Then any Hermitian matrix can be written in the form: H = X (⃗ y j ,⃗ v j ,⃗ w j )∈∆( Y,V,W) α j n O i=1 (σ z ) y ji σ + v ji σ − w ji + X (⃗ y j ,⃗ v j ,⃗ w j )∈∆( Y,V,W) α † j n O i=1 (σ z ) y ji σ + w ji σ − v ji , (2.6) where Y = {⃗ y 1 ,...,⃗ y r } with ⃗ y i ∈ {0,1} n , V = {⃗ v 1 ,...,⃗ v r } with ⃗ v i ∈ {0,1} n , and W = {⃗ w 1 ,..., ⃗ w r } with ⃗ w i ∈{0,1} n are such that the corresponding α i ̸= 0. Here ∆( Y,V,W) = {(⃗ y i ,⃗ v i , ⃗ w i )|y i ∈ Y,v i ∈ V,w i ∈ W}, where ∆ simply takes any indexed element sets and creates the set of the index-wise confederated tuples. A tuple (⃗ y i ,⃗ v i , ⃗ w i ) specifies the indices inwhichwechoseσ z ,σ + ,orσ − foreachnonzeroterm. Oncethatchoiceismade,hermiticity demands the corresponding second term seen in Eq. 2.6 to be part of the Hamiltonian as 17 well. However, there are restrictions on what vectors can be chosen. Specifically, ⃗ y i · ⃗ w i =0 and ⃗ y i · ⃗ v i = 0, since choosing σ z and a σ ± would actually mean selecting σ ± with a new coefficient − α j instead. Likewise, it should also be the case that ⃗ v i · ⃗ w i = 0 – otherwise the term would be equivalent to two terms with the same coefficient halved and one taking the term σ z , the other taking the identity term. As such, this added constraints on ∆( Y,V,W) so that the representations are unique, which must be enforced before applying the theorem below because the uniqueness of the basis will be actively used. As an example, consider the driver Hamiltonian discussed in the previous section: H d = n− 1 X i=1 (σ x i σ x i+1 +σ y i σ y i+1 )=2 n− 1 X i=1 (σ + i σ − i+1 +σ − i σ + i+1 ) (2.7) For this Hamiltonian, Y ={ ⃗ 0,..., ⃗ 0},V ={⃗ e 1 ,...,⃗ e n− 1 },W ={⃗ e 2 ,...,⃗ e n } - where ⃗ e i refers to the standard basis vectors. While the notation is somewhat awkward, it becomes useful for expressing our first major result: Theorem 2.3.1 (Algebraic Condition for Commutativity). A Hermitian Matrix H com- mutes with a linear constraint C if and only if ⃗ c· (⃗ v j − ⃗ w j )=0 for all ⃗ v j , ⃗ w j ∈∆( V,W). 18 Proof. Using the form for H we introduced earlier, we can see that: h H, ˆ C i = X (⃗ y j ,⃗ v j ,⃗ w j )∈∆( Y,V,W) " α j n O i=1 (σ z ) y ji σ + v ji σ − w ji , n X k=1 c k σ z k # + X (⃗ y j ,⃗ v j ,⃗ w j )∈∆( Y,V,W) " α † j n O i=1 (σ z ) y ji σ + w ji σ − v ji , n X k=1 c k σ z k # = X (⃗ y j ,⃗ v j ,⃗ w j )∈∆( Y,V,W) 2α j n X k=1 c k (v jk − w jk ) ! n O i=1 (σ z ) y ji σ + v ji σ − w ji + X (⃗ y j ,⃗ v j ,⃗ w j )∈∆( Y,V,W) 2α † j n X k=1 c k (w jk − v jk ) ! n O i=1 (σ z ) y ji σ + w ji σ − v ji = X (⃗ y j ,⃗ v j ,⃗ w j )∈∆( Y,V,W) 2α j ⃗ c· (⃗ v j − ⃗ w j ) n O i=1 (σ z ) y ji σ + v ji σ − w ji − X (⃗ y j ,⃗ v j ,⃗ w j )∈∆( Y,V,W) 2α † j ⃗ c· (⃗ v j − ⃗ w j ) n O i=1 (σ z ) y ji σ + w ji σ − v ji (2.8) Since the set defined by the tuples {(⃗ y j ,⃗ v j , ⃗ w j )} is a linearly independent set Eq. 3.6 =0 iff ⃗ c· (⃗ v j − ⃗ w j )=0 for all j. Consider the example discussed in Section 5.2 in the context of Theorem 2.3.1. It is easy to see that v 1 = (1,0,0,0),w 1 = (0,1,0,0),v 2 = (0,1,0,0),w 2 = (0,0,1,0),v 3 = (0,0,1,0),w 3 =(0,0,0,1) would satisfy Eq. 3.6, defining H d = P 3 i=1 σ + i σ − i+1 +σ − i σ + i+1 . Note that more vector pairs will also satisfy the condition, for example v 4 = (0,0,0,1) and w 4 = (1,0,0,0). 2.4 Bounded weight drivers The algebraic condition of Theorem 2.3.1 can be used as a starting point to understand severalfeaturesofCQA.Oneofthemismotivatedbythefactthatactualimplementationsof quantumannealingwilllikelyimposeaboundontheweightofthedriveroperatorsavailable. 19 Hence, given a set of linear constraints, we could restrict our search for commuting drivers to those with bounded weight. Consider a set C = {C 1 ,...,C m } of linear constraints on n variables, and let C M be the m× n matrix with coefficients c ij (recall C i (x) = P j c ij x j ). Then, Theorem 2.3.1 tells us that there is a non-diagonal driver commuting with all the constraints, if and only if, the linear system C M ⃗ u = ⃗ 0, where ⃗ u =⃗ v− ⃗ w∈{− 1,0,1} n . Furthermore, we can see that the numberofcomponentsof⃗ uthatarenon-zeroisalowerboundfortheweightofsuchadriver (the weight could be higher if we allow σ z operators acting on the variables associated with the vanishing components of ⃗ u). This leads to a simpler analysis of the case of bounded weight drivers. Assumethatweareonlyallowedweightk drivers,andwefurtherimposethecondition thattheyareconstructedfromk1-qubitoperatorsthatactnontriviallyinthecomputational basis (in our case, it means these are chosen from {σ + ,σ − }). Then, the number of such operatorsis2 k− 1 n k . Forfixed k,thisispolynomialinn,andsoitistractabletocheckallthe possiblevectors⃗ utofindwhichonessatisfythecondition C M ⃗ u=0. Fromthosethatdo, we can construct the corresponding weight k driver that will commute with all the constraints, by assigning σ + to qubit i if u i = 1, and σ − if u i = − 1. This is a simple but very useful result in practice: brute force searching for solutions to the condition from Theorem 2.3.1 findsallpossibledriverHamiltoniansthatcommutewiththeembeddedconstraintoperators up to a certain weight in time polynomial on the system size. The simplest and most relevant case for practical applications is that of 2-local oper- ators, since 2-body interactions are more easily engineered than higher order ones. In this case the condition of Theorem 2.3.1 implies an even simpler characterization of when a set of constraints allow for commuting drivers. Corollary 2.4.0.1. Let C ={C 1 ,...,C m } be a set of linear constraints and C M the asso- ciated matrix of coefficients. Then a 2-local driver that commutes with all the constraints exists, if and only if, the matrix C M has a pair of columns that are either equal, or opposite. 20 2-local means that ⃗ u in the condition C M ⃗ u = 0 has only 2 non zero components, and we can take these to be (1,1) or (1,− 1), since the other two possibilities would produce the corresponding Hermitian conjugates. Since multiplying a matrix by a column vector results in a linear combination of the columns of the matrix, with the coefficients given by the vector components, the condition C M ⃗ u = 0 will state that C M has two columns that are the same (if the non zero components of ⃗ u are (1,− 1)), or are opposites (if they are (1,1)). To any distinct pair of columns that satisfy one of these conditions we can associate a distinct weight-2 driver that commutes with the constraints, so the maximum number of such possible weight-2 drivers is n 2 . 2.5 The Problem ILP-QCOMMUTE Having found a simple algebraic condition for expressing the commutation relationship of any Hermitian matrix with a linear constraint, we wish to exploit that fact here to find what the general complexity of knowing the existence of such a Hermitian matrix is. We consider the following problem: Definition (ILP-QCOMMUTE). Given a set C ={C 1 ,...,C m } of linear constraints such that ˆ C i = P n j=1 c ij σ z j , over a space C 2 n with c ij ∈ Z, is there a Hermitian Matrix H, with O(poly(n)) nonzero coefficients over a basis {χ 1 ,χ 2 ,χ 3 ,χ 4 } N n , such that h H, ˆ C i i =0 for all ˆ C i and H has at least one off-diagonal term in the spin-z basis? SolvingthisproblemwouldbeusefulforconstructingHamiltoniandriversforquantum annealing. We can also define 0-1-LP-QCOMMUTE as the binary version, where c ij ∈{0,1}, and also{-1,0,1}-LP-QCOMMUTE, where c ij ∈{− 1,0,1}, the type of coefficients used when representing problems like 1-in-3 3-SAT as an ILP. One of the central results of this chapter is that these problems are NP-Complete [1, 47], which can be shown by reducing them to the EQUAL SUBSET SUM problem [2]. This reduction is simple and straightforward for ILP-QCOMMUTE, and we discuss it in this section to give a sense of the connection between 21 the two problems. However, this proof is not enough to imply that 0-1-LP-QCOMMUTE and {-1,0,1}-LP-QCOMMUTE are also NP-Complete, since they could both very well be easier subclasses of ILP-QCOMMUTE. That is not the case, but the proof for 0-1-LP-QCOMMUTE is more involved (and rather tedious) and thus is presented in the Appendix ??. Definition (EQUAL SUBSET SUM). Given a set S = {s 1 ,s 2 ,...,s n }, with s i ∈ Z + , are there two non-empty disjoint subsets, A,B such that P a i ∈A a i = P b i ∈B b i ? The EQUAL SUBSET SUM problem is known to be NP-Complete [48]. We map an in- stance of the EQUAL SUBSET SUM problem to the ILP-QCOMMUTE problem; the former defined over a set S = {s 1 ,s 2 ,...,s n }, with s i ∈ Z + . Consider the constraint operator defined by ˆ C = P n i=1 s i σ z i , and the vector ⃗ s = (s 1 ,...,s n ). Suppose we can find vectors ⃗ v, ⃗ w with binary components, such that ⃗ s· (⃗ v− ⃗ w) = 0 (the algebraic condition derived in Theorem 2.3.1). Then the indices corresponding to the nonzero components of ⃗ v and ⃗ w can be used to identify the sets A and B (respectively) in the EQUAL SUBSET SUM problem. Suppose there is a solution H to ILP-QCOMMUTE. From Theorem 2.3.1, it follows that ⃗ c i · (⃗ v− ⃗ w) = 0 for the vector ⃗ c associated with constraint C and any nonzero term in the basis {1,σ z ,σ + ,σ − } ⊗ n with a term σ ± on at least one qubit will be enough to define a new H ′ that will be associated with a solution to EQUAL SUBSET SUM. At least one such elementexistsforH becauseH hasatleastoneoff-diagonalterminthespin-zbasis. Wecan associateanyoff-diagonaltermofHwith ⃗ y,⃗ v, ⃗ wsuchthatH ′ =α N n i=1 (σ z ) y i (σ + ) v i (σ − ) w i + α † N n i=1 (σ z ) y i (σ + ) w i (σ − ) v i is a matrix with only that off-diagonal term and its complex conjugate for some α and⃗ v̸= ⃗ 0, ⃗ w̸= ⃗ 0. Then for a specific off-diagonal term, every non-zero entry in ⃗ v between 1 and n, call it i, picks an integer s i ∈ S for the set A, and ⃗ w does likewise for the set B, providing a solution to the corresponding instance of EQUAL SUBSET SUM since ⃗ s· (⃗ v− ⃗ w) = P s i ∈A s i − P s i ∈B s i = 0. Suppose there is a solution A,B to EQUAL SUBSET SUM, then define ⃗ v such that v i = 1 (w i = 1) if and only if s i is in A (B). Then H = N n i=1 (σ + ) v i (σ − ) w i + N n i=1 (σ + ) w i (σ − ) v i is a solution to ILP-QCOMMUTE since P s i ∈A s i + P s i ∈B s i =⃗ s· (⃗ v− ⃗ w)=0. Hence, we have the following result. 22 Theorem 2.5.1. ILP-QCOMMUTE is NP-Hard. Given this result, we can show NP-Completeness by noting that we can check for an off-diagonal term in the spin-z basis in polynomial time. Let H be a proposed solution to the ILP-QCOMMUTE such that there exists nonequivalent indices i,j such that entry h ij ̸= 0. Checking that H commutes with the constraints is polynomial time. For k ∈ {1,...,n}, check if Tr k Hσ + k ̸= 0 or Tr k Hσ − k ̸= 0. H has at least one element h ij that is off- diagonal in the spin-z basis if and only if there exists k such that at least one of these terms is nonzero. Since the partial trace of tensor products is the trace over a specific tensor component, this can be done quickly. Corollary 2.5.1.1. ILP-QCOMMUTE is NP-Complete. While we have shown that this problem is NP-Hard, we note that for any specific instance of the problem, the practical runtime can still be tractable and therefore a useful avenue for even the hardest instances of optimization problems.Also take note that since everyunitary operatoristheexponentialofacorrespondingHermitian operator,knowingthe existenceofaunitaryoperatorthatcommuteswiththeconstraintoperatorsisparamountto knowing a Hermitian matrix exists with the same property. As such, our result immediately translates to the QAOA setting where one wishes to construct unitary operators that will commute with the embedded linear constraint operators. 2.5.1 Bounded Weight ILP-QCOMMUTE Despite the NP-hardness of ILP-QCOMMUTE, Section 2.4 discusses a simple polynomial time algorithm to find driver terms up to some weight k. Consider this modified version of ILP-QCOMMUTE, which asks about the existence of a Hermitian matrix that commutes with the constraints, but consists of interaction terms up to weight k. Definition (ILP-QCOMMUTE-k-LOCAL). Given a set C = {C 1 ,...,C m } of linear con- straints such that ˆ C i = P n j=1 c ij σ z j , over a space C 2 n with c ij ∈ Z, is there a Hermitian 23 Matrix H, withO(poly(n)) nonzero coefficients over a basis {χ 1 ,χ 2 ,χ 3 ,χ 4 } N n and no term withweighthigherthank, suchthat h H, ˆ C i i =0forall ˆ C i andH hasatleastoneoff-diagonal term in the spin-z basis? Theorem 2.5.2. ILP-QCOMMUTE-k-LOCAL is in P for k inO(1). Proof. Apply the brute force approach described in Section 2.4. Since k ∈O(1), the algo- rithm runs in time n O(1) . This shows that for a practical application where the Hamiltonian driver should be all local, we can tractably find such a Hamiltonian driver. Moreover, any H that is all local and commutes with the constraints can be constructed by placing the right coefficients on the found terms by brute forcing the expression in Theorem 2.3.1 (they form a basis for all Hamiltonians that commute with the constraints up to weight k). 2.6 Reachability within the Feasible Space Intheprevioussection,weprovedthatfindingaHermitianmatrixwhichcommuteswith acollectionoflinearspin-zconstraintsisNP-Complete. Therelatedquestionbecomesfinding a Hermitian matrix which commutes with the constraints, but also connects the feasibility space. Note that two states |p⟩,|q⟩ are connected if they are in the same commutation subspace of H, that is⟨p|H r |q⟩̸=0for some r∈Z + . As such, for anypair of solutions i,j in the feasibility space, their associated vectors in the computational basis |i⟩,|j⟩ should be in the same commutation subspace. In general, when finding a driver Hamiltonian for an anneal that should solve an optimization task, we wish to find a driver that satisfies this condition so that we can ensure thatitwillbeabletoreachtheentirefeasibilityspace, sincecommutingwiththeconstraints alone is not enough to ensure this will happen. Consider the graph partitioning example discussedinSection5.2andSection2.5, clearlyσ + 1 σ − 2 +σ − 1 σ + 2 commuteswiththeconstraint, 24 but fails to connect the entire feasible space. For example, the state|0011⟩ is disconnected from|1100⟩ under this driver Hamiltonian. While it succeeds not to mix solution states to nonsolution states, it does not mix all solution states with each other. Weintroducetheproblem ILP-QCOMMUTE-NONTRIVIALwhichaskstofindadriverterm that not only commutes with the constraints, but acts nontrivially on the feasible space, so that the action of the driver is such that there exists one solution state which the driver maps to another solution state in the feasible space. If we think about the solution states as verticesinagraph,thentransitionsinducedbydrivertermsaretheedges(andadriverterm can induce more than one such edge). Then the problem ILP-QCOMMUTE-NONTRIVIAL asks that the driver term found induces at least one edge in the graph of feasible solutions. For thegraphpartitioningexamplediscussedabove,thetermwediscussedisclearlyasolutionto the problem ILP-QCOMMUTE-NONTRIVIAL since it connects|1010⟩ to|0110⟩, both of which are in the feasible space for this example. Let P b i i be the projection operator corresponding to the energy eigenvalue b i for the constraint operator ˆ C i . Then this formally defines the problem ILP-QCOMMUTE-NONTRIVIAL: Definition (ILP-QCOMMUTE-NONTRIVIAL). Given a set C = {C 1 ,...,C m } of linear constraints and constraint values b = {b 1 ,...,b m } such that ˆ C i = P n j=1 c ij σ z j over a space C 2 n with c ij ∈Z, is there a Hermitian Matrix H, with O(poly(n)) nonzero coefficients over a basis {χ 1 ,χ 2 ,χ 3 ,χ 4 } N n , such that h H, ˆ C i i = 0 for all ˆ C i and P b 1 1 ··· P bm m HP bm m ··· P b 1 1 has at least one off-diagonal term in the spin-z basis? The main difference is that while ILP-QCOMMUTE required nontrivial off-diagonal terms in the spin-z basis, ILP-QCOMMUTE-NONTRIVIAL specifically requires these to be nontrivial in the constraint space of interest, which in general is a non-polynomial problem to verify (i.e. knowing a Hamiltonian has or fails to have an eigenvector for a specific energy level would allow one to know if a Hamiltonian has a solution to hard problems). We show that this problem is at least NP-Hard by reducing a problem closely related to EQUAL SUBSET SUM to ILP-QCOMMUTE-NONTRIVIAL. We begin with the famous NP-Complete SUBSET SUM problem: 25 Definition (SUBSET SUM). Given a set S ={s 1 ,...,s n } of integers and an integer target value T, is there a subset S 1 such that P s∈S 1 s=T? While SUBSET SUM asks about the existence of a single solution, we are interested in at least two solutions, defining the problem: Definition (2-OR-MORE SUBSET SUM). Given a set S ={s 1 ,...,s n } and a target value T, are there two subsets S 1 ,S 2 such that P s∈S 1 s= P s∈S 2 s=T? WeshowthatlikeSUBSET SUM(overpositiveintegers[3]), 2-OR-MORE SUBSET SUMisalso NP-Hard: Lemma 2.6.1. 2-OR-MORE SUBSET SUM is NP-Hard. Proof. Consider an instance of the SUBSET SUM problem with a set S ={s 1 ,...,s n } and a target value T such that s i > 0 for all s i . We construct a new instance of the 2-OR-MORE SUBSET SUM problem with set S ′ ={s 1 ,...,s n ,T} and the same target value T. First we show how to relate a solution to the original SUBSET SUM instance from a solution to the constructed 2-OR-MORE SUBSET SUM problem. Let S 1 ,S 2 be solutions to the new 2-OR-MORE SUBSET SUMinstance,eitheroneornoneofthesolutionsusestheelementT. If neither does, either one is a solution to the instance of the original SUBSET SUM problem. Without loss of generality, suppose S 1 uses the value T, then S 1 cannot use any other value since every other value is greater than zero, hence S 1 ={T}. Since S 2 cannot use any value other than T if T ∈S 2 and S 1 ̸=S 2 , it follows that T / ∈S 2 . Then S 2 ⊆ S and P s∈S 2 s=T. We now show how to relate a solution to the constructed 2-OR-MORE SUBSET SUM instance given a solution to the SUBSET SUM instance. Let S 1 be a solution to the SUBSET SUM problem. Then S 1 ,{T} is a solution to the 2-OR-MORE SUBSET SUM instance. Since SUBSET SUM is NP-Hard over positive integers, 2-OR-MORE SUBSET SUM is NP- Hard as well. 26 2-OR-MORE SUBSET SUM is closely related to EQUAL SUBSET SUM because both ask about the existence of two subsets with equal sums, but 2-OR-MORE SUBSET SUM adds the further constraint that these two subsets should have a specific sum. We show that ILP-QCOMMUTE-NONTRIVIAL is NP-Hard through a reduction to 2-OR-MORE SUBSET SUM. Note again that ILP-QCOMMUTE-NONTRIVIAL is not verifiable in polynomial time [49–51] and so this reduction is only for the decision version of the problem 2-OR-MORE SUBSET SUM. Theorem 2.6.2. ILP-QCOMMUTE-NONTRIVIAL is NP-Hard. Proof. Consider an instance of the 2-OR-MORE SUBSET SUM problem with a set S = {s 1 ,...,s n } and a target value T. Define the constraint operator ˆ S = P n j=1 s j σ z j and a target energy value P n j=1 s j − 2T. Suppose this instance of ILP-QCOMMUTE-NONTRIVIAL has a solution. Then there are at least two eigenvectors|⃗ v⟩,| ⃗ w⟩ of ˆ S with eigenvalue P n j=1 s j − 2T such that the two eigenvalues can be written in the spin-z basis with ⃗ v, ⃗ w∈{0,1} n . Like with ILP-QCOMMUTE and EQUAL SUBSET SUM, the nonzero elements of⃗ v and ⃗ w describe two sets S 1 ,S 2 such that s i ∈ S 1 (s i ∈ S 2 ) if and only if v i = 1 (w i = 1). Since ˆ S|⃗ v⟩ = P n j=1 s j (1− 2v j ) |⃗ v⟩ = P n j=1 s j − 2 P n j=1 s j v j |⃗ v⟩, it follows that P n j=1 s j v j =T. The same logic works for ⃗ w and so P s∈S 1 s = P s∈S 2 s = T. Then 2-OR-MORE SUBSET SUM must have a solution as well, specifically S 1 ,S 2 . Suppose 2-OR-MORE SUBSET SUM has a solution. Then there are two nonequal subsets S 1 ,S 2 ofS suchthat P s i ∈S 1 s i = P s i ∈S 2 s 2 =T. Thenlet⃗ v =(v 1 ,...,v n )(⃗ w =(w 1 ,...,w n )) with v i =1 (w i =1) if s i ∈S 1 (s i ∈S 2 ). 27 Then: ˆ S|⃗ v⟩= n X j=1 s j (1− 2v j ) ! |⃗ v⟩ (2.9) = n X j=1 s j ! − 2 n X j=1 s j v j ! |⃗ v⟩ (2.10) = n X j=1 s j ! − 2T ! |⃗ v⟩. (2.11) Thesamelogicworksfor| ⃗ w⟩andso|⃗ v⟩,| ⃗ w⟩arebotheigenvectorsof ˆ S witheigenvalue P n j=1 s j − 2T. Then|⃗ v⟩⟨ ⃗ w|+| ⃗ w⟩⟨⃗ v| is a driver term that nontrivially maps solution states of this constraint problem to one another. In practical applications, we are often able to quickly find some driver terms that commute with the constraints, but then need to know whether they are sufficient to connect the entire feasible space. This raises the following question: given k driver terms (individual basis terms) that commute with the constraints, does some linear combination of them with nonzero coefficients connect the entire feasibility space? In other words, given that we have found k driver terms that commute with the constraints, can we guarantee that some linear combinationofthemwillhavethewholefeasiblesubspaceasitssmallestinvariantsubspace? Note that not every driver term that does commute with a subspace is necessary to solve this problem. For example, in the case of the constraint ˆ C = P n i σ z i , it suffices to use the driver terms σ + i σ − i+1 +σ − i σ + i+1 for i ∈ [n− 1]. Any linear combination with nonzero coefficients, H d = P n− 1 i λ i σ + i σ − i+1 +σ − i σ + i+1 , then is a valid Hamiltonian driver to connect the feasible space. Then an extra term, like σ + 1 σ − 3 + σ − 1 σ + 3 is unnecessary, because if |ϕ ⟩ is in the constrained subspace, it follows that (σ + 1 σ − 2 + σ − 1 σ + 2 )|ϕ ⟩, (σ + 2 σ − 3 + σ − 2 σ + 3 )|ϕ ⟩, (σ + 1 σ − 2 + σ − 1 σ + 2 )(σ + 2 σ − 3 +σ − 2 σ + 3 )|ϕ ⟩, and (σ + 2 σ − 3 +σ − 2 σ + 3 )(σ + 1 σ − 2 +σ − 1 σ + 2 )|ϕ ⟩ are as well. Note that 28 (σ + 1 σ − 3 +σ − 1 σ + 3 ) = (σ + 1 σ − 2 +σ − 1 σ + 2 )(σ + 2 σ − 3 +σ − 2 σ + 3 )+(σ + 2 σ − 3 +σ − 2 σ + 3 )(σ + 1 σ − 2 +σ − 1 σ + 2 ). If a Hermitian matrix M can be decomposed into a linear combination of products of operators chosen from the set of driver terms{ ˆ G k }, and|ϕ ⟩ is a state in the constrained space, then for any state|ψ ⟩ such that⟨ψ |M|ϕ ⟩̸= 0 (i.e., any state reachable from|ϕ ⟩ through the action of M), is also reachable through the action of the driver terms in { ˆ G k } for the state |ψ ⟩. The set of Hamiltonians that commute with a given set of constraints form an algebra (known as the commutant of the set of constraints). Each one of the driver terms we are considering can be seen as a generator of this commutant algebra. This leads us to define the problem ILP-QIRREDUCIBLECOMMUTE-GIVEN-k formally: Definition (ILP-QIRREDUCIBLECOMMUTE-GIVEN-k). Given a set C ={C 1 ,...,C m } of linear constraints and constraint values b ={b 1 ,...,b m } such that ˆ C i = P n j=1 c ij σ z j over a space C 2 n with c ij ∈ Z, and a set of basis terms G = { ˆ G 1 ,..., ˆ G k } such that ˆ G i ∈ {χ 1 ,χ 2 ,χ 3 ,χ 4 } N n , doesG connect the entire nonzero eigenspace of the operator P b 1 1 ··· P bm m ? As such, ILP-QIRREDUCIBLECOMMUTE-GIVEN-k asks if a given set of driver terms is able to connect the entire feasible space of a set of constraints with the given constraint values. We show that this problem is also NP-Hard by reducing ILP-QCOMMUTE-NONTRIVIAL to ILP-QIRREDUCIBLECOMMUTE-GIVEN-k. We do so by finding a mapping for any instance of ILP-QCOMMUTE-NONTRIVIAL to an instance of ILP-QIRREDUCIBLECOMMUTE-GIVEN-k. Con- sider such an instance with constraints {C 1 ,...,C m } and constraint values {b 1 ,...,b m }. Find an integer a 1 such that∥⃗ c i ∥ 1 ∥⃗ c i ∥ 1 + P i− 1 j=1 a j and ˆ G i = (σ + n+2i− 1 σ − n+2i + σ − n+2i− 1 σ + n+2i ), 1 ≤ i ≤ k. By this construction, when restricted to the ancilla variables, the feasible subspace is spanned by the vectors{ N k i=1 |i 1 i 2 ⟩,i 1 +i 2 =1}. Given any element in this subspace, the action of the ˆ G i driver terms is sufficient to guarantee that any other element of this subspace can also be generated. To proceed with the reduction, we then give the constraints{F 1 ,...,F m } with constraint values{b 1 + P k i=1 a i ,...,b m + P k i=1 a i } re- spectively and the drivers { ˆ G 1 ,..., ˆ G k } to our ILP-QIRREDUCIBLECOMMUTE-GIVEN-k solver oracle. Since our k driver terms are all over the added qubits x n+1 to x n+2k , it should be clear that these driver terms say nothing about the feasibility space of the original problem over qubits x 1 to x n . Suppose we are told that our k drivers are sufficient, i.e., they can generate the whole feasible subspace by acting on any one element of that subspace. Then clearly there are no driver terms for the original ILP-QCOMMUTE-NONTRIVIAL decision problem, since none of the k driver terms operate over qubits associated with variables x 1 to x n . Likewise if we are told our k drivers are not sufficient, then clearly there must be at least one nontrivial driver for the original problem, since the drivers are enough to generate all elements of the feasible subspace when restricted to the ancillas. Note that this solves ILP-QCOMMUTE-NONTRIVIAL without giving us a token to verify it. Because this is the most general unstructured version 30 oftheproblem,isitpossiblethatadifferentcomplexityresultcanbefoundforamorestruc- tured questioning of the same problem. We note that the problem can also have a stronger complexityresult,suchasarelationshiptoahigherclassinthepolynomialhierarchy[52–54], like #P[55], to which it has some natural analogues. Given this result and the result of Section 2.4, we can often find drivers that satisfy the condition stated in Theorem 2.3.1, but may not connect the entire feasible space. Still, there remain many avenues for exploiting such terms; for example, alongside the ordinary transverse field such that universality is maintained, but with biasing towards a subspace of the solution space. This gives us a way to adjust the knob of using higher order terms when the ordinary transverse field struggles to find a solution. Such driver terms can also be beneficialforexploringnewsolutionsusingreverseannealing[56, 57], especiallyforsolutions that are higher hamming distance away since the transverse field generally struggles to find such solutions. Anotherwaytoleverageourresultistobruteforcetheproblemforasetofconstraints over a small enough subspace that it becomes polynomially tractable. Over the other vari- ables, we apply the usual transverse field and enforce the other constraints as penalty terms in the final Hamiltonian. These approaches can also be adopted to the constraints that are geometrically local (like in a two dimensional grid). 2.7 Conclusion In this chapter, we addressed the computational complexity of finding driver Hamil- tonians for quantum annealing processes which aim at solving optimization or feasibility problems with several linear constraints. We develop a simple and intuitive algebraic frame- work for understanding whether a Hamiltonian commutes with a set of constraints or not. While this result is interesting mathematically in its own right, we mainly focus on the 31 problem posed in Ref. [20] about algorithmically finding driver Hamiltonians for optimiza- tion problems with several linear constraints. Most significantly, the condition is useful for finding a reduction of the NP-Hard problem EQUAL SUBSET SUMS to finding such a driver Hamiltonian, thereby allowing us to categorize the complexity of this problem. We also showed that ILP-QCOMMUTE-NONTRIVIAL and ILP-QIRREDUCIBLECOMMUTE-GIVEN-k are at least NP-Hard. But these problems could well be in a higher complexity class in the polynomial hierarchy - like #P, to which ILP-QIRREDUCIBLECOMMUTE-GIVEN-k has some similarity. However, for most common implementations the Hamiltonians are of bounded weight, and the relevant complexity class ILP-QCOMMUTE-k-LOCAL for a small integer k is in P. Hence, there is a simple brute force algorithm, as detailed in Section 2.4, to find a basis for all possible driver Hamiltonians of this bounded locality. However, the results from ILP-QIRREDUCIBLECOMMUTE-GIVEN-k say that given a set of driver terms, it is intractable to know whether the found basis can sustain a Hamiltonian that connects the entire feasibility space for the linear constraints. As such, we present a polynomial time algorithm that is guaranteed to find a basis for all possible Hamiltonians that commute with a set of embedded constraint operators up to a certain weight, but with no guarantees that the found Hamiltonian is able to connect the entire feasibility space that the constraints specified. However, for some important problems it is actually possible to exploit the constraint structure to guarantee that driver terms with low weight will be sufficient to reach all feasible states. This is the case, for example, for the graph coloring problem discussed in Section 5.2 and Ref. [20]. Our result also applies to finding mixing operators for Quantum Alternating Operator Ansatz (QAOA) [24, 29]. To implement highly nontrivial driver Hamiltonians for an anneal, it also becomes necessary to find a new initial Hamiltonian that is then evolved slowly with a simple linear interpolation to the driver Hamiltonian since thermal equilibration to the 32 driver Hamiltonian may be difficult. It then becomes relevant how can we construct such a Hamiltonian for a given driver Hamiltonian, such that we can guarantee that we reach the right constrained space. This is a fundamental question for future research. While we have shown these problems to be NP-Hard, we have not shown what the average hardness of this class is or what the typical hardness is for instances of interest for specific applications. Especially pertinent become sets of instances in which the practical runtime for finding driver Hamiltonians remains tractable or the hardness of the problem comes from having to search a large feasible space for an optimum solution rather than pinpointing a very small feasible space. It is also interesting to note that our algebraic formulation is agnostic to the stoquas- ticity of the terms found. In the presented basis of Section 2.3, the stoquasticity of the individual basis terms, as written in Eq. 2.6, is determined by the amplitudes α and its conjugate pair α † . Commutation is invariant under altering α (α † will adjust as we alter α to keep the term pair commutative). Once we have found driver terms that are suitable for a problem, it then raises the question of what effect, if any, choosing coefficients that will make them stoquastic or non-stoquastic will have on the anneal [58–61]. This is another direction that requires further study. 2.8 Acknowledgements for Chapter The research is based upon work (partially) supported by the Office of the Director ofNationalIntelligence(ODNI),IntelligenceAdvancedResearchProjectsActivity(IARPA) andtheDefenseAdvancedResearchProjectsAgency(DARPA),viatheU.S.ArmyResearch Office contract W911NF-17-C-0050. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies orendorsements,eitherexpressedorimplied,oftheODNI,IARPA,ortheU.S.Government. 33 The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon. 34 Chapter 3 Constructing Driver Hamiltonians and Mixing Operators for Optimization Problems with General Constraints In the previous chapter, we found a formula for reasoning about imposing linear con- straint symmetries on quantum operators. In this chapter, we find a more general formula that allows us to reason about imposing general (rather than linear) constraint symmetries on quantum operators such that the quantum evolution is guaranteed to remain in one of the constraint subspaces if the initial state only has support within this subspace. The ideas expressed within this chapter are the result of collaboration from the author with Federico M. Spedalieri, Stuart A. Hadfield, and Eleanor Rieffel. This thesis is the first publication to bear them, but they are subject to publication in another medium at a later date. 3.1 Introduction Quantum annealing and QAOA remain some of the most popular approaches for uti- lizing quantum systems to solve optimization problems. Recent developments in both have focused on finding specialized drivers (in CQA) and tailored ans¨ atze (in QAOA) to better exploit the underlying classical symmetries of the specific optimization problem one tries to solve. In this chapter we give a general formula for reasoning about imposing several 35 constraints on the driver Hamiltonians (in CQA) or the mixing operators (in QAOA). Re- sults in the previous chapter (and Ref. [34]) are sufficient to show that the decision problem of knowing whether a Hamiltonian (or a quantum operator) commutes with a set of con- straints is NP-Hard, but left open the possibility that the local Hamiltonian (or quantum operator) version is not in P. That is, there is a polynomial time algorithm shown in the previous chapter for finding such a local Hamiltonian (or recognizing that none exists) for linear constraints. In this chapter, by utilizing a more general algebraic condition, we are able to delineate a polynomial time algorithm for finding local operators that commute with a set of general constraints. 3.2 AnAlgebraicConditionforCommutingwithConstraints Given any complete basis for a single qubit system, we can extend that basis to define a basis over n qubits. Doing this with the found eigenmatrices of Chapter 2 defines a basis {1,σ z ,σ + ,σ − } ⊗ n . Note again that α j σ ± i † =α † j σ ∓ i for α j ∈C. Then any Hermitian matrix can be written in the form: H = X (⃗ y j ,⃗ v j ,⃗ w j )∈∆( Y,V,W) α j n O i=1 (σ z ) y ji σ + v ji σ − w ji + X (⃗ y j ,⃗ v j ,⃗ w j )∈∆( Y,V,W) α † j n O i=1 (σ z ) y ji σ + w ji σ − v ji , (3.1) Refer to Chapter 2 for definitions and necessary conditions on the terms appearing in this formula. Recall the main algebraic result from Chapter 2: Theorem 3.2.1 (Algebraic Condition for Commutativity). A Hermitian Matrix H com- mutes with an embedding of a linear constraint C if and only if ⃗ c· (⃗ v j − ⃗ w j ) = 0 for all ⃗ v j , ⃗ w j ∈∆( V,W). 36 In this chapter, we wish to find similar algebraic conditions for more general constraints. For higher term constraints, we can write the constraint in the general form: ˆ C = X J K ∈J J K Y k∈K σ z k , (3.2) WhereJ is a collection of sets of polynomial size in the number of qubits n, each with a coefficient J K and a set of qubit indices K⊆ [1,...,n]. Notice that σ + σ z =− σ z σ + =σ + and σ − σ z =− σ z σ − =− σ − . Using the form for H from Eq. 3.1, define T(⃗ y,⃗ v, ⃗ w) = Q n i=1 (σ z i ) y i σ + i v i σ − i w i such that T † (⃗ y,⃗ v, ⃗ w)=T(⃗ y, ⃗ w,⃗ v). ThenconsiderthecommutationofageneralH withtheconstraintembeddingoperator for a general constraint C: h H, ˆ C i = X (⃗ y j ,⃗ v j ,⃗ w j )∈∆( Y,V,W) " α j n Y i=1 (σ z i ) y ji σ + i v ji σ − i w ji , X J K ∈J J K Y k∈K σ z k # + X (⃗ y j ,⃗ v j ,⃗ w j )∈∆( Y,V,W) " α † j n Y i=1 (σ z i ) y ji σ + i w ji σ − i v ji , X J K ∈J J K Y k∈K σ z k # = X (⃗ y j ,⃗ v j ,⃗ w j )∈∆( Y,V,W) α j X J K ∈J J K L jK + X (⃗ y j ,⃗ v j ,⃗ w j )∈∆( Y,V,W) α † j X J K ∈J J K R jK (3.3) 37 where: L jK = Y k/ ∈K (σ z k ) y jk σ + k w jk σ − k v jk !" Y k∈K (σ z k ) y jk σ + k w jk σ − k v jk , Y k∈k σ z k # = Y k/ ∈K (σ z k ) y jk σ + k w jk σ − k v jk ! Y k∈K (σ z k ) y jk σ + k w jk σ − k v jk σ z k − Y k∈K σ z k (σ z k ) y jk σ + k w jk σ − k v jk ! = Y k/ ∈K (σ z k ) y jk σ + k w jk σ − k v jk ! Y k∈K (− 1) v jk (σ z k ) y jk +µ jk σ + k w jk σ − k v jk − Y k/ ∈K (σ z k ) y jk σ + k w jk σ − k v jk ! Y k∈K (− 1) w jk (σ z k ) y jk +µ jk σ + k w jk σ − k v jk = Y k∈K (− 1) v jk − (− 1) w jk ! Y k∈K (σ z k ) µ jk T(⃗ y j ,⃗ v j , ⃗ w j ) =2 |K| Y k∈K v jk − w jk ! T(⃗ y j + ⃗ µ j ,⃗ v j , ⃗ w j ) (3.4) with µ jk = (1− v jk )(1− w jk ) (identity or spin-z were placed on location k by the j-th basis term), and so ⃗ µ j =(µ j1 ,...,µ jn ). Likewise, it can be shown: R jK =2 |K| Y k∈K w jk − v jk ! T(⃗ y j + ⃗ µ j , ⃗ w j ,⃗ v j ) (3.5) and so: h H, ˆ C i = X (⃗ y j ,⃗ v j ,⃗ w j )∈∆( Y,V,W) α j X J K ∈J J K 2 |K| Y k∈K v jk − w jk ! T(⃗ y j + ⃗ µ j ,⃗ v j , ⃗ w j ) + X (⃗ y j ,⃗ v j ,⃗ w j )∈∆( Y,V,W) α † j X J K ∈J J K 2 |K| Y k∈K w jk − v jk ! T(⃗ y j + ⃗ µ j , ⃗ w j ,⃗ v j ) = X (⃗ y j ,⃗ v j ,⃗ w j )∈∆( Y,V,W) α j X J K ∈J J K 2 |K| Y k∈K u jk ! T(⃗ y j + ⃗ µ j ,⃗ v j , ⃗ w j ) + X (⃗ y j ,⃗ v j ,⃗ w j )∈∆( Y,V,W) α † j X J K ∈J J K 2 |K| Y k∈K − u jk ! T(⃗ y j + ⃗ µ j , ⃗ w j ,⃗ v j ). (3.6) 38 Define v(J,Y,V,W)∈C 2 3n , which appears like Eq. 3.6 except T(⃗ y,⃗ v, ⃗ w) is replaced by ⃗ y⊗ ⃗ v⊗ ⃗ w. ClearlyEq.3.6iszeroifandonlyifv(J,Y,V,W)=0. Wecandefine u j = ⃗ v j − ⃗ w j since ⃗ v j , ⃗ w j never match on an index. GiventhatH hasO(poly(n))nonzerotermsoverthebasis,thereareatmostO(|J|poly(n)) such terms to check, all of which must be zero. 3.3 Sufficient Condition for Quadratic Constraints A class of constraints of particular interest are quadratic constraints, which can be used to describe a large class of optimization problems. Certain NP-Hard problems, for example, are more naturally described as quadratic constraints. In this section we give a sufficient condition for quadratic constraints that generalizes naturally from Theorem 3.2.1. Let us check that Eq. 3.6 matches our result from Theorem 3.2.1 when considering linear constraints, i.e. ˆ C = P n k=1 J k σ z k : h H, ˆ C i = X (⃗ y j ,⃗ v j ,⃗ w j )∈∆( Y,V,W) α j X J K ∈J J K 2 |K| Y k∈K v jk − w jk ! T(⃗ y j + ⃗ µ j ,⃗ v j , ⃗ w j ) + X (⃗ y j ,⃗ v j ,⃗ w j )∈∆( Y,V,W) α † j X J K ∈J J K 2 |K| Y k∈K w jk − v jk ! T(⃗ y j + ⃗ µ j , ⃗ w j ,⃗ v j ) = 2 X (⃗ y j ,⃗ v j ,⃗ w j )∈∆( Y,V,W) α j n X k=1 J k (v jk − w jk )T(⃗ y j , ⃗ w j ,⃗ v j ) +2 X (⃗ y j ,⃗ v j ,⃗ w j )∈∆( Y,V,W) α † j n X k=1 J k (w jk − v jk )T(⃗ y j , ⃗ w j ,⃗ v j ) = 2 X (⃗ y j ,⃗ v j ,⃗ w j )∈∆( Y,V,W) α j ⃗ J· (⃗ v j − ⃗ w j )T(⃗ y j , ⃗ w j ,⃗ v j ) − 2 X (⃗ y j ,⃗ v j ,⃗ w j )∈∆( Y,V,W) α † j ⃗ J· (⃗ v j − ⃗ w j )T(⃗ y j , ⃗ w j ,⃗ v j ) (3.7) 39 Now, let us consider the case with quadratic and linear terms in the constraint. For such a constraint, the constraint embedding operator has the form ˆ C = P i h i σ z i + P ij J ij σ z i σ z j and so: h H, ˆ C i = X (⃗ y j ,⃗ v j ,⃗ w j )∈∆( Y,V,W) α j X J K ∈J J K 2 |K| Y k∈K v jk − w jk ! T(⃗ y j + ⃗ µ j ,⃗ v j , ⃗ w j ) + X (⃗ y j ,⃗ v j ,⃗ w j )∈∆( Y,V,W) α † j X J K ∈J J K 2 |K| Y k∈K w jk − v jk ! T(⃗ y j + ⃗ µ j , ⃗ w j ,⃗ v j ) = 2 X (⃗ y j ,⃗ v j ,⃗ w j )∈∆( Y,V,W) α j n X k=1 h k (v jk − w jk )T(⃗ y j , ⃗ w j ,⃗ v j ) +2 X (⃗ y j ,⃗ v j ,⃗ w j )∈∆( Y,V,W) α † j n X k=1 h k (w jk − v jk )T(⃗ y j , ⃗ w j ,⃗ v j ) +4 X (⃗ y j ,⃗ v j ,⃗ w j )∈∆( Y,V,W) α j X kl J kl (v jk − w jk )(v jl − w jl )(σ z k ) µ jk (σ z l ) µ jl T(⃗ y j , ⃗ w j ,⃗ v j ) +4 X (⃗ y j ,⃗ v j ,⃗ w j )∈∆( Y,V,W) α † j n X kl J kl (w jk − v jk )(w jl − v jl )(σ z k ) µ jk (σ z l ) µ jl T(⃗ y j , ⃗ w j ,⃗ v j ) (3.8) To get a sufficient condition, assume terms associated with {h i } and {J ij } are zero re- spectively. The first condition is the previous result ⃗ h· (⃗ v j − ⃗ w j ) = ⃗ h· ⃗ u j = 0. A sufficient condition on the second is P kl J kl (v jk − w jk )(v jl − w jl )= P kl J kl u jk u jl = ⃗ u j T · J M · ⃗ u j =0. Theorem 3.3.1 (Sufficient Condition for Quadratic Constraints) . A Hermitian Matrix H commutes with an embedding of a quadratic constraint C if ⃗ h· ⃗ u j = 0 and ⃗ u T j · J M · ⃗ u j = 0 for all ⃗ u j . Whilethisconditionissufficient,itisnotnecessaryfor H tocommutewiththeconstraint embedding operator. We consider a simple concrete counterexample. Consider constraints associated with the maximum independent set problem, where we wish to maximize a set of 40 vertices S⊆ V such that no v∈S has an edge to any other w∈S. This can be represented with the quadratic constraints: ∀v∈V, X (v,w)∈E x v x w =0. (3.9) Clearly the Hamiltonian (σ + v +σ − v ) Q (v,w)∈E (1+σ z w ) commutes with the embedded con- straint, but does not satisfy this sufficient condition. 3.4 ImposingConstraintSymmetriesonQuantumGeneral and Bounded Weight Operators In this section we turn our attention to the related computational tasks associated with finding a Hamiltonian that commutes with an arbitrary set of constraints. The definitions follow those discussed Chapter 2 using the notation introduced in Sec. 3.2. Definition (CP-QCOMMUTE). Given a set C = {C 1 ,...,C m } of constraints such that ˆ C i = P K∈J J K Q n k∈K σ z j , over a space C 2 n with J K ∈ Z, each K ⊆ [1,...,n], and |J| = O(poly(n)), is there a Hermitian Matrix H, with O(poly(n)) nonzero coefficients over a basis{χ 1 ,χ 2 ,χ 3 ,χ 4 } N n , such that h H, ˆ C i i =0 for all ˆ C i and H has at least one off-diagonal term in the spin-z basis? Theorem 3.4.1. CP-QCOMMUTE is NP-Hard. Proof. Follows immediately from the hardness of ILP-QCOMMUTE. Theorem 3.4.2. CP-QCOMMUTE is NP-Complete. Proof. Note,asinthepreviousChapter,itispossibletoverifyifasolutionH hasatleastone off-diagonalbasisterminpolynomialtime. InSec.3.2weshowedthatcheckingcommutation can be done in polynomial time as well. 41 Notice that, as in the previous chapter, the relationship between unitary operators and Hamiltoniansmeansthat theseresultstranslatedirectly to thesituationof designing mixing operators for Quantum Alternating Operator Ansatz algorithms. Definition (CP-QCOMMUTE-k-LOCAL). Given a set C = {C 1 ,...,C m } of constraints such that ˆ C i = P K∈J J K Q n k∈K σ z j , over a space C 2 n with J K ∈ Z, is there a Hermitian Matrix H, with O(poly(n)) nonzero coefficients over a basis {χ 1 ,χ 2 ,χ 3 ,χ 4 } N n , such that h H, ˆ C i i =0 for all ˆ C i and H has at least one off-diagonal term in the spin-z basis? We gave an interesting formula for finding a sufficient condition for quadratic con- straints, but it is not clear that it can be exploited to find a polynomial time algorithm for local operators as occurred in the previous chapter. This leaves the interesting ques- tion whether CP-QCOMMUTE-k-LOCAL (or related classes such as QP-QCOMMUTE-k-LOCAL for quadratic terms) are in P like ILP-QCOMMUTE-k-LOCAL. 3.5 Conclusion Inthischapter,wegaveanalgebraicconditionforfindingcommutingHamiltonianswith embedded general constraints. This formula can be useful for constructing driver Hamilto- nians for this much more general class of classical optimization problems as well as native examples of exploring invariance under multibody spin-z interactions (for example in quan- tumchemistry). Weshowthat CP-QCOMMUTEisNP-Complete. Wepresentasimplesufficient condition in Theorem 3.3.1 for quadratic constraints that can be useful for quadratic pro- gramming problems. In Chapter 2, we showed that a 2-local Hamiltonian nonviable for certain classes of constraints. The algebraic condition found in this Chapter may be utilized to find similar exclusionary statements for problems using more general constraints, thereby showing that certain constructions are not viable under certain resource restrictions. 42 3.6 Acknowledgements for Chapter Results in this chapter arose from discussions held with Federico M. Spedalieri, Eleanor Rieffel, and Stuart A. Hadfield. We thank sponsorship from NASA Ames and the Feynman Academy program at USRA to facilitate these discussions. We are grateful for the support from the NASA Ames Research Center and from DARPA under IAA 8839, Annex 128. The research is based upon work (partially) supported by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA) andtheDefenseAdvancedResearchProjectsAgency(DARPA),viatheU.S.ArmyResearch Office contract W911NF-17-C-0050. The author was also supported by the USRA Feynman Quantum Academy and funded by the NAMS R&D Student Program under contract no. NNA16BD14C. 43 Chapter 4 Quantum Annealing with Special Drivers for Circuit Fault Diagnostics In this chapter, we consider the task of defining a CQA protocol to solve the combina- tionalcircuitfaultdiagnosticproblem. ThischapterisbasedonworkinRef.[23]. Giventhe combinatorial problem underlying circuit fault diagnostics and the typical low connectivity andrelativesparsityofproblemsinpracticemakethistaskanaturalcandidateforexploring quantum algorithms that can work on NISQ devices. The circuit fault diagnostic problem is very structured and in this chapter as well as Chapter 5 we tailor approaches to exploit these symmetries. Along the way we illustrate important and fundamental questions for CQA protocols, such as selecting a reasonable annealing schedule, understanding the behav- ior of the minimum gap, and exploring how the degeneracy and fault count of the solution space affects a typical CQA protocol. 4.1 Introduction Quantum computation can offer new and exciting prospects for solving problems that are difficult with existing classical approaches [62, 63]. Quantum Annealing (QA) and the Quantum Approximate Optimization Algorithm [24] are two popular approaches for doing quantum computation on near-term quantum devices [64–67]. While Driver Hamiltonians 44 in QA and ansatz strategies in QAOA are often selected based on their simplicity to be implementedonnear-termdevices, anexcitingprospecttoimprovetheperformanceofthese systems is to tailor these constructions to the underlying structure of the problem being solved [6, 33, 68]. Many modern quantum computational frameworks attempt to leverage quantumphenomenatosolvecombinatorialoptimizationproblems. Amongtheseisquantum annealing (QA), which leverages adiabatic (or diabatic) evolution from one Hamiltonian to another,usuallythroughastandardinterpolation. Applicationsincombinationalcircuitfault diagnostics (CCFD) have gained considerable attention [4, 32, 33] for quantum annealers to potentially leverage quantum speed up. Developments in Constrained Quantum Annealing (CQA) [19, 20, 34] and Quantum Alternating Operator Ansatz (QAOA) [27, 29, 31, 69, 70] have centered around more novel driver operators, with higher body interaction terms than the transverse field. As noted in Ref. [33], driver Hamiltonians beyond the transverse field are pertinent to understanding what QA can bring to solving such problems and this work can be seen as a step in this direction. In this chapter we delineate a CQA approach for solving single input/output CCFD problems that maintains support only on the space of valid diagnoses. By constraining the anneal to only explore feasible states associated with a given CCFD instance, we find a square root reduction in the size of the explored space compared to the standard transverse field approach in the explicit mapping setting [4]. It also avoids the need to impose energy penaltiesonthefinalHamiltoniantosuppressinvaliddiagnoses, thatinturnresultsinstrin- gent requirements for the range of the interaction parameters. The driver terms that are needed to do this are all local (with bounded weight), representing the possible transforma- tions that takes a feasible configuration to another feasible configuration involving only the variables associated with a single circuit gate. These driver terms are sufficient and minimal for describing a driver Hamiltonian that constrains the evolution to the subspace of all valid diagnoses, such that they form a generating set for all possible driver Hamiltonians that constrain the evolution to this feasibility space [34]. 45 To solve an instance of the CCFD problem we need, (i) an initial 1-local Hamiltonian that has as its ground state the trivial diagnosis (all faults occur in the output gates), (ii) a driver Hamiltonian composed of the driver terms discussed in the previous paragraph, and (iii)afinalHamiltonian(also1-local)thatfavorsdiagnoseswiththesmallernumberoffaults. Utilizing a standard interpolation between these three Hamiltonians, the wavefunction of the system begins with support on a specific valid configuration and then explores all valid configurationswhiletheevolutionoftheinstantaneousHamiltoniantothefinalHamiltonian eventuallyrestrictsthewavefunctiontosupportonthesubsetofdiagnoseswiththeminimum number of faults. The chapter is organized as follows. In Section 5.3, we provide background on the combinatorial problem CCFD and in Section 4.3 delineate how our approach can be utilized tosolvethisproblem. InSec.4.4,weanalyzetheminimumgapoftheannealingspectrumfor asetofrandomlygeneratednon-degenerateinstancesofamodifiedversionoftheC17circuit form the ISCAS benchmarks [71], as well as the success probability for different choices of the annealing schedule. We then construct a larger, richer family of instances on which we run a generic but improved annealing schedule, and study the relationship between the success probability, the degeneracy of the ground space of the instance, and the number of faults of the minimum fault diagnosis. While these instances require 52 qubits to represent explicitly, because the closed system evolution occurs in a much smaller space, we are still able to tractably simulate these systems on a workstation. The results suggest our approach can be beneficial for employing quantum annealers to solve instances where higher number of faults are needed to be able to diagnose the circuit. 4.2 CombinatorialOptimizationforCircuitFaultDiagnostics Consider a combinational circuit with N wires (or locations), with N I inputs and N O outputs. Each wire can take one of two values, x i = 0 or x i = 1, so a state of the circuit 46 0 0 1 1 0 0 0 1 1 1 1 1 1 1 1 0 0 Figure 4.1: A schematic of the C17 circuit with a fault on output of the third NAND gate for a given input/output pair. In Sec. 4.4, we generate new random circuits by replacing each NAND gate with a randomly selected two input logic gate. is associated with an N bit string or assignment, (x 1 ,...,x N ). The wires are connected to logical gates in such a way that any wire that is not an input or an output for the entire circuit, is connected to exactly two gates (it is the output of one gate and the input to another one). Inputs (outputs) to the circuit are inputs (outputs) to (for) one gate. The connectivityofthegatesdeterminesthefunctionperformedbythecircuit. Thisisreferredto asthestructuraldescriptionofthecircuit,andisoneofthemostcommonlyusedabstractions in testing digital systems [72]. For the purpose of our discussion here, we consider gates that are either fan out gates (FAN), which maps a single input to two outputs such that the two outputs have values equivalent to the input, an inverter gate (INV) or one of the following logic gates: OR, AND, XOR, NOR, NAND. The methods described in this chapter can be extended trivially to a more general k 1 − inputs, k 2 − outputs gate. Since we are using the structural description of the circuit, we will use the most well known structural fault model, the stuck-at fault (SAF) model of CCFD [72]. In this model, 47 a wire is either healthy, or fails by being either stuck-at-0 (SA0) or stuck-at-1 (SA1). Given aninput-outputpair, andadescriptionofthecircuitstructure, adiagnosisisanexplanation of this input-output relationship that is consistent with the chosen fault model. To describe a diagnosis, we will add to the N bits describing the values of each wire, another N fault bits f i , associated with each of the wires, that would indicate if a wire is faulty (f i = 1) or not (f i =0). Under the assumption that every wire is equally likely to be faulty, the goal of the circuit fault diagnosis (CFD) problem is to find the minimal number of faults necessary to provide a consistent explanation for the outputs of the circuit [32]. We will refer to any diagnosis with the minimum number of faults as a minimum fault diagnosis (MFD). Note thatourrepresentationdoesnotspecifythetypeoffault(SA0orSA1), butthisinformation can be inferred from the state of the wire: if a wire has a value x i = 1 (x i = 0) and f i = 1, we can conclude the wire must be SA1 (SA0). 4.2.1 The set of valid diagnoses Wecanrefertoany2N− bitstringasadiagnosis,inthesensethatitprovidesadescrip- tionofthestateofthewiresonthesystemandthelocationofpotentialfaults. However,not any 2N− bit string will represent a fault assignment that is consistent with the chosen fault model. We will call this subset of consistent assignments, the set of valid diagnoses. Hence, a valid diagnosis can be represented by a 2N− bit string, (x 1 ,...,x n ,f 1 ,...,f n ), where the first N bits describe the state of the circuit, while the last N bits flag whether a wire is healthy or not. For every wire location i, if the assignment value x i is not compatible with the logic of the gate that it is an output to and the corresponding inputs for that gate, then the fault flag f i must be set to 1 for the assignment to valid, otherwise the assignment is invalid (as a false negative). Likewise, if the assignment value x i is compatible, then the fault flag f i must be set to 0, otherwise the assignment is invalid (as a false positive). 48 The last N O wires correspond to the output bits of the circuit and are fixed by the problem instance. Given a collection of empirical outputs for a faulty circuit, there are potentially many valid diagnoses to explain this phenomenon, since the faults could occur anywhere in the circuit, but a fault occurring on a particular wire does not mean the sub- sequent wire values are faulty since they were compatible with their respective gates (i.e. the inputs for the gate were different from what the non-faulty circuit would have and this led to their erroneous assignment). The first N I wires correspond to the inputs and are in principle fixed. However, we must allow for a fault to occur before the input wires reach the first gates, so the first N I wires are allowed to take any value: any discrepancy with the actual inputs to the circuit will be reflected by the corresponding fault bit being set to 1. The consistency of the fault assignment is imposed on the wires and fault bits associated with every gate in the circuit. We can illustrate this with the following example. Consider the bit string (x i 1 ,x i 2 ,x o ;f i 1 ,f i 2 ,f o ) representing the wires and fault bits associated with a NAND gate in the circuit. The configuration (110;000) represents a consistent assignment, in the sense that the output of the gate is compatible with its inputs, i.e., x o = NAND(x i 1 ,x i 2 ), and no wire is flagged as faulty; (100;001) is also a consistent assignment, because even though the output wire is inconsistent with the inputs and the gatefunction(x o ̸=NAND(x i 1 ,x i 2 )), theoutputwireisflaggedasfaulty( f o =1). Examples of inconsistent assignments would be (111;000) and (011;001), where in the first case the output is incorrect but its fault bit indicates no fault, while in the second case the output is correct but the fault bit is flagging it as faulty. Note that the fault bit of a gate input is of no importance in deciding whether the output of that gate is faulty or not: the output is faulty (healthy) if it is inconsistent (consistent) with the values of the inputs of the gate that produces it. A valid diagnosis will then be described by a 2N− bit string, where for all circuit gates, the corresponding wires and fault bits form a consistent assignment for the specific function of that gate. In Table 5.1 we present an exhaustive list of all consistent assignments for a NAND gate, indexed by the values of the input and output wires. 49 (x i 1 ,x i 2 ;x o ) Consistent assignments (x i 1 ,x i 2 ,x o ;f i 1 ,f i 2 ,f o ) (0,0;0) (0,0,0;0,0,1), (0,0,0;1,0,1), (0,0,0;0,1,1), (0,0,0;1,1,1) (1,0;0) (1,0,0;0,0,1), (1,0,0;1,0,1), (1,0,0;0,1,1), (1,0,0;1,1,1) (0,1;0) (0,1,0;0,0,1), (0,1,0;1,0,1), (0,1,0;0,1,1), (0,1,0;1,1,1) (1,1;0) (1,1,0;0,0,0), (1,1,0;1,0,0), (1,1,0;0,1,0), (1,1,0;1,1,0) (0,0;1) (0,0,1;0,0,0), (0,0,1;1,0,0), (0,0,1;0,1,0), (0,0,1;1,1,0) (1,0;1) (1,0,1;0,0,0), (1,0,1;1,0,0), (1,0,1;0,1,0), (1,0,1;1,1,0) (0,1;1) (0,1,1;0,0,0), (0,1,1;1,0,0), (0,1,1;0,1,0), (0,1,1;1,1,0) (1,1;1) (1,1,1;0,0,1), (1,1,1;1,0,1), (1,1,1;0,1,1), (1,1,1;1,1,1) Table 4.1: Consistent assignments for a NAND gate: note that there is no restriction on the values of the inputs, their fault bits and the output (hence, we have 2 5 =32 configurations). However, the output fault bit is completely determined by the values of the inputs and the output. We can now see how to construct any possible valid diagnosis. Given any arbitrary valueforthefirst N− N O wires,andthevaluesoftheN O outputwires,wecanthenunivocally set the value of each fault bit f i such that it is consistent with the value of the wire and the constraint imposed by the preceding gate. Hence the set of valid diagnoses has 2 N− N O elements. 4.3 Constrained Quantum Annealing Approach Quantum Annealing (QA) has been used in many combinatorial optimization tasks [9, 73–75]. Many studied quantum annealing protocols involve two Hamiltonians: a driver Hamiltonian like the transverse field H d =− P i σ x i and a final Hamiltonian made of single body and two body spin-z interaction terms H f = P i h i σ z i + P j J ij σ z i σ z j [4] (σ x i and σ z i are the standard Pauli matrices). A typical linear transverse field annealing schedule will then be H(s(t)) = (1− s(t))H d + s(t)H f with the time function s(t) = t/T f for some specified total time T f . The transverse field driver Hamiltonian mixes configurations that 50 differ by a single bit flip. Applying it to our CFD problem would result in generating configurations that are not valid diagnoses. One way to deal with this issue is to add a term to the final Hamiltonian that penalizes any configuration that is not a valid diagnosis. This is a commonly used technique to deal with constraints both in classical and quantum optimization [46, 76]. In QA this introduces other problems, such as a determining the right energy scale for these penalty terms, and requiring more qubit connectivity to represent them. Constrained Quantum Annealing (CQA) [19, 20, 34] and the Quantum Alternating Operator Ansatz (QAOA) [27, 29, 31, 69, 70] have been introduced to avoid these precise issues, and our work in this chapter aligns with the spirit of these approaches. In the case of CQA, a ground state for the driver Hamiltonian can be more dif- ficult to prepare and so annealing protocols are often generalized beyond a two Hamil- tonian interpolation. To overcome this barrier, we consider annealing protocols involv- ing three Hamiltonians, a form for annealing schedules often seen when exploiting ‘cata- lysts’ [22, 77, 78]: H(s(t)) = A(s(t))H i +B(s(t))H d +C(s(t))H f with s : [0,T f ]→ [0,1], A(s(0))=1− B(s(0))=1− C(s(0))=1, and A(s(T f ))=B(s(T f ))=1− C(s(T f ))=0. In Sec.4.3.1weshowhowtoexplicitlyconstructthedriverHamiltonianforthisapproachandin Sec. 4.3.2 describe a general CQA protocol using an interpolation of an initial Hamiltonian, the driver Hamiltonian, and a final Hamiltonian. 4.3.1 Special Drivers for CQA Solvers tackling CFD ThespecialCFDdriversaredesignedtolocallyconnectpairsofvaliddiagnoses. Toeach gate in the circuit we will associate operators that are non-diagonal in the computational basis, which couple two valid diagnoses that differ only on the qubits associated with that specific gate. Since each input and output wire of a gate is mapped to two qubits (one for the wire value and one for the fault bit), the special drivers will have a weight that is double the number of inputs plus outputs of the gate. 51 We will now show how to construct the required drivers for a NAND gate (this pro- cess can be extended to any other gate). Assume that we have a valid diagnosis for the whole circuit, and let |x i 1 x i 2 x o ;f i 1 f i 2 f o ⟩ be the corresponding state restricted to the qubits associated with this specific NAND gate. We want to find other configurations of these 6 qubits that also represent a valid diagnosis. The transformed configuration must be locally and globally consistent with the assignment of other wires and fault bits in the circuit. To ensure global consistency, we must (i) keep the value of the output wire fixed, to prevent propagating its effect forward, and (ii) flip the wire and associated fault bit of any input simultaneously, sincethiswillkeepthenewassignmentconsistentwithanypreviousgatefor which this input wire is an output. Summarizing, we can only change the input wires and their fault bits (as long as the wire and the corresponding fault bit are flipped together), and the output fault bit to ensure the local consistency of the new diagnosis. To illustrate the idea, consider a valid diagnosis whose state restricted to the qubits representing a given NAND gate are given by the state|100;001⟩, where the output wire is flagged as faulty because it does not respect the gate’s functionality. If we flip x i 1 and f i 1 , we get |000;101⟩ and the output fault bit remains unchanged since this configuration still represents a failing NAND gate. Same if we flip both inputs and their fault bits. However, if we flip only x i 2 and f i 2 , the resulting configuration |110;011⟩ has an output wire that now is consistent with the NAND gate, but its fault bit marks it as faulty. Hence, we should also flip the output fault bit to 0 to make it consistent, resulting in the new local configuration |110;010⟩. Gatheringallthisinformation,weseethattherequireddrivertermstoimplement these local transformations of a valid diagnosis will be: {|000;101⟩⟨100;001|+h.c.,|110;010⟩⟨100;001|+h.c.,|010;111⟩⟨100;001|+h.c.}, (4.1) where h.c. stands for Hermitian conjugate. The same analysis can be carried out for any valid configuration of the NAND gate. There are 32 such valid configurations for a 2-input, 52 1-output gate (cfr. Table 5.1). For each one of those we need 3 driver terms (per valid configuration) to perform the input flips (and flip the output bit if necessary). But due to the hermiticity requirement, the total number of driver terms is reduced by half, to 48. 4.3.2 Implementing CQA protocols for CFD with special drivers In order to map an instance of CFD to our quantum annealing framework, we need to specify an initial Hamiltonian, a driver Hamiltonian and a final Hamiltonian. The driver Hamiltonian can be constructed from the circuit description, as a linear combination of the drivers corresponding to each gate in the gate set G of the circuit. There is actually somefreedomonwhatthecoefficientsofthislinearcombinationcouldbe(aslongastheyare nonzero)[34]. Thiscouldpotentiallyimpacttheperformanceoftheapproachanddeveloping useful strategies for choosing these coefficients is an intriguing direction for future research. Here we will discuss this briefly later when we compare the effects of stoquastic versus non- stoquastic drivers, but for now we will proceed with the following (stoquastic) assignment H D =− X g∈G H g , (4.2) where each H g is a sum of the drivers associated with the gate g, which we described how to construct in the previous section (see Eq. 4.1). Each gate has a bounded number of drivers associated with it, so the total number of terms in H D scales linearly with the number of gates in the circuit. TheinitialHamiltonianshouldhaveaground state,writteninthecomputationalbasis, that is a valid diagnosis. Such a state can always be found by simply assuming that all gates are healthy except for the gates that produce the output bits that are known to be wrong. The input and output wires are fixed by the instance, and the interior wire values are computed by propagating the input through the healthy circuit. The fault bits are set to 0, except for the ones corresponding to the faulty output wires, which are set to 1. 53 Let |x 1 ...x N f 1 ...f N ⟩ be that initial state. Then we can initialize the system by implementing an initial Hamiltonian diagonal in the computational basis given by H I = N X i=1 ((− 1) x i σ z i +(− 1) f i σ z N+i ), (4.3) whichhasthisinitialstateasitsgroundstate. ThefinalHamiltonianmustpenalizediagnoses that have a higher number of faults, so the Hamiltonian H F =− 2N X i=N+1 σ z i , (4.4) where the sum is only over the fault bit qubits. We can then combine all this into a time dependent Hamiltonian: H(s(t))=A(s(t))H I +B(s(t))H D +C(s(t))H F . (4.5) Hence, the system starts in a specific valid diagnosis, and once the driver is turned on it will explore only the space of valid diagnoses. Finally, at the end, the final Hamiltonian will favor diagnoses with the lowest number of faults. In comparison to the transverse field approach used with explicit mappings in Refs. [4, 32, 33], there is no need for penalty terms associated with constraining the solution to be a valid diagnosis, which also requires setting a coefficient in the order of N O for the most general type of instances. The action of the driver Hamiltonian allows us to reach any valid diagnosis from any other valid diagnosis, so the adiabatic evolution generates a superposition of all valid di- agnoses. The graph induced by H D in this space is connected and regular, with degree (3G 2 +G 1 ), where G 2 and G 1 are the number of gates with 2 and 1 inputs, respectively. Furthermore, for a given connectivity, the corresponding induced graphs are isomorphic for all instances (i.e., any input-output pairs). This is due to the fact that the driver terms are essentially just performing all possible flips of the inputs of a gate, while the fault bits are 54 flipped accordingly to enforce the constraints of a valid diagnoses. In the next Section, we benchmark the CQA approach discussed on small sized circuits. 4.4 BenchmarksonSyntheticInstances: SpectralGaps and Simulated Annealing Schedules 4.4.1 Generalized ISCAS C17 Random Circuits: Spectral Gaps and Parameterized Annealing Schedule OurfirstexampleisacircuitknownasC17(seeFig.4.1),thatappearedintheISCAS’85 benchmarks [71, 79]. This circuit has 17 wires, 5 inputs, 2 outputs, 6 NAND gates and 3 FAN gates. Using our approach, it requires 34 qubits to represent all the wire values and the fault bits. The reduced dynamics occurs on a space of dimension 2 15 , so it is possible to simulate the closed system evolution and compute the spectrum along the anneal. WegeneratednewcircuitinstancesbyrandomlyreplacingtheNANDgateswithother 2-input, 1 output gates, randomly chosen from the set {NAND, AND, OR, NOR, XOR}. Then we randomly chose the 5 input bits, simulated the circuit to compute the 2 output bits, and then flipped the value of these 2 outputs. The CFD instance is then defined by the circuit description, and the input-output pair. Note that the number of output bits that are faulty is always an upper bound on the number of faults in the minimal fault diagnosis (MFD), since assigning the fault only to those output bits is itself a valid diagnosis. That is also the reason why we flipped both output bits, to make room for a non-trivial MFD with a single fault. 4.4.1.1 Spectral Gaps for C17 Instances Fig. 4.2.1 shows the energy gap between the ground state and the first excited state at every point during the anneal, for 100 randomly generated instances. We can see that the 55 (1) (2) Figure4.2: (1)showstheinstantaneousminimumgapfor100randominstancesofthegener- alizedC17circuitatregulardiscretizedpointsoftime. (2)similarlyshowstheinstantaneous ground state gap for the same instances with a non-stoquastic version of the driver Hamil- tonian. 56 gap always starts at 4 (due to the form of the initial Hamiltonian, and the fact that any two valid diagnoses are at least Hamming distance 2 away). Depending on the degeneracy of the instance, the gap either has an absolute minimum during the anneal (non-degenerate), or goes to 0 at the end of the anneal (degenerate). An interesting feature is that the minimum gap for non-degenerate instances seems to be localized on the last third of the anneal. The generic shape of the gap as a function of the anneal parameter could be exploited to evolve the system faster at the beginning, when the gap is large. This would only result in a constant factor reduction in the total annealing time required, so it has no impact on the complexity, but is useful for simulating larger circuits. We also computed the spectrum corresponding to flipping the sign of the Hamiltonian driver, which produces a non-stoquastic Hamiltonian. Due to the infamous sign problem, non-stoquastic Hamiltonians are thought to be harder to simulate than stoquastic ones [77, 80], leading some to conjecture they may provide some computational speedup in some situ- ations and non-stoquastic catalyst Hamiltonians, such as considered in Ref. [77], are able to achieve large improvements over catalyst-free approaches in the finite and infinite [21] range ferromagneticp-spinmodel. Ref.[81]likewiseshowspossibleimprovementsfromintroducing non-stoquastic terms in a driver Hamiltonian. However, there is also evidence [61] that they can produce smaller energy gaps, resulting in a computational slowdown. In Fig. 4.2.2 we show the minimum gap for the non-stoquastic version of the driver Hamiltonian. Comparing this with Fig. 4.2.1 we can see that in this case, the minimum gap seems to be generically smaller, and the region where the gap is small is also larger. It is important to point out that these are not the only choices for the coefficients of the terms in the driver Hamiltonian. Since these coefficients determine how different diagnoses are mixed by the driver, one could imagine that a better choice, perhaps one that exploits the particular structure of the circuit, may favor some sort of interference that could enhance diagnoses with lower number of faults and result in a computational advantage. However, 57 it is not clear at this point if that is feasible, or even what the complexity of finding those coefficient would be. It is nonetheless an intriguing direction for future research. 4.4.1.2 Choosing annealing schedule to exploit spectrum features Forourannealingschedule,wechosefunctionsA(s(t))=1− s 2 (t),B(s(t))=4s(1− s), and C(s(t)) = s 2 (t) (the instantaneous Hamiltonian is then given by Eq. 4.5). We utilized thegenericfeaturesoftheenergygapforC17(andinSec.4.4.2forC26),togenerateasingle parameter annealing schedule, that evolves faster at the beginning of the anneal (when the gapislarger), andthenslowsdownwhentraversingtheregioncontainingtheminimumgap. For a given amount of total anneal time T f , the annealing schedule assigns T 0 (T 0 < T f ) units of time for the range [0,s 0 ], following a quadratic function, and then shifts at point T 0 to a linear function over the interval [s 0 ,1] such that s ′ (T 0 ) is not a point of discontinuity. These conditions yield: s(t)= s 0(T f /T 0)− 1 T 0(T 0 − T f) t 2 + 1− s 0(2(T f /T 0)− 1) T 0 − T f t, t<T 0 s 0 − 1 T 0 − T f t+ 1− s 0(T f /T 0) 1− (T f /T 0) , t>T 0 In the experiments on C17, T 0 was set to 20 units and s 0 was set to 3/4. We used the DifferentialEquations.jl [82] package in Julia [83] to simulate the Schr¨ odinger equation for the time interval [0,T] in this and the succeeding Section. Results in Fig. 4.3 suggests that while this parameterized function is not an optimal strategy (a strategy that optimally exploits the adiabatic condition is represented in the plot by opt adia), it has a clear advan- tage over the linear function and is therefore used in the next Section on benchmarks for the C26 circuit. 58 (1) (2) Figure 4.3: (1) and (2) show the success probability versus the minimum gap for 22 non- degenerate single fault instances of the C17, with 40 units and 80 units of time respectively, for the single parameter function (labeled param), the linear function (labeled linear) and a piece-wise linear function fitted over 100 evenly spaced points, such that the slope is proportional to the instantaneous inverse gap squared between evenly spaced s i− 1 and s i (labeled opt adia). 59 Figure4.4: AschematicoftheC26circuit. ForourC26benchmarks,wereplaceeachNAND gate with a randomly selected two input logic gate as well as randomly select inputs to the circuit. Each has 8 2-input logic gates, 6 FAN gates, 26 wires, 6 inputs, and 4 outputs. 4.4.2 Generalized C26 Random Circuits: Effects of Degeneracy and Multi-Fault Solution Spaces The number of outputs of a circuit is an upper bound on the number of faults of the minimum fault diagnosis. Since the C17 has only two outputs, if both of these are faulty, the MFD can have either 1 or 2 faults; the latter case is trivial (just assign the faults to the output bits), and the non-trivial case of a single fault is rather simple. In order to explore instances that offer higher complexity, we need to consider circuits with a larger number of outputs. In this section, we consider a modification of C17 to a 26 wire, 6 input, 4 output circuit that we call C26 (see Fig. 4.4 for a schematic). While the circuits studied in Sec. 4.4.1 required 34 qubits to represent explicitly, the evolution of the wavefunction is restricted to a subspace of dimension 2 15 . In this section we consider a larger class of circuits that require 52 qubits to represent explicitly, but whose wavefunction evolution is restricted to a subspace of size 2 22 , which is still simulatable for sparse Hamiltonians. 60 (1) (2) Figure 4.5: (1) shows a box and whisker plot of the logarithm of the minimum gap as a function of the number of faults in the MFD, while (2) shows a box and whisker plot of the location of the minimum gap versus the MFD number of faults, for randomly generated instances of C26 with non-degenerate ground states. We generated random instances of C26 following the same procedure described in Sec. 4.4.1, and we selected 100 non-degenerate instances for each value (1, 2 and 3) of the number of faults in the MFD. We selected non-degenerate instances so that we could identifythelocationandsizeoftheminimumgap(degenerateinstanceshaveavanishinggap as s→ 1), and also because these are in some sense the hardest instances. In Fig. 4.5.1 we see that, even though the distribution of the minimum gap seems to spread towards smaller values as we increase the number of faults in the MFD, the median shows only a rather mild dependence. This means that an annealing schedule designed to solve, say, all non-outlier 1-fault MFD instances, will also solve about 75% and just above 50% of 2-fault and 3-fault MFD instances, respectively. In Fig. 4.5.2, we also see that the location of the gap (similar to the C17 case) occurs in a well defined range for most instances and this appears stable with regard to the number of faults of the MFD. However, as we consider larger circuits, the degeneracy of the MFD subspace will also potentiallyincrease. Thatcouldbebeneficialforaquantumannealingapproach,sincejumps from the ground state to excited states may still result in optimal solutions at the end of the anneal. Of course, we cannot know a priori whether a given instance is degenerate or 61 non-degenerate, and what is the number of faults of its MFD. In order to gain some more insight into the performance of our approach on generic CFD instances, we simulated the closed system evolution and computed the success probability for a new set of randomly generated instances of CFD for C26, without implementing any type of post-selection. The only feature we exploited was the generic form of the spectrum (as seen in Fig. 4.2.1), and implemented the modified annealing schedule detailed in Sec. 4.4.1. We simulated the annealing protocol for 100 randomly generated instances (including those with degenerate ground spaces) for a total annealing time of 40, 60, 80, and 140 units. Note that (following our parameterized annealing schedule) 20, 40, 60, and 120 time units were consumed on the interval [3/4,1], using a linear function, and always 20 time units were used on the interval [0,3/4] using a quadratic function. Fig. 4.6 shows the results of theseexperiments. First, we note thatrandominstanceseem morelikelytohave2-faultand 3-fault MFDs, and these in turn tend to be more degenerate. Instances with a 1-fault MFD were always non-degenerate, and were the hardest instances to solve. In general, we can see that instances with low degeneracy are typically harder than those with high degeneracy. From a quantum annealing point of view we can understand this as results of the many jumps the system must go through to go from the ground state to the first state that is not partofthegroundstatemanifold(i.e., thesetofeigenstatesthatwillendupconverginginto the ground state of the final Hamiltonian). High degeneracy seems to add extra protection against the effects of diabatic transitions, resulting in higher success probabilities. 4.5 Conclusions In this chapter, we introduced a novel approach to solving the CFD problem, where the quantum state throughout the anneal is constrained to the space of valid diagnoses. of size 2 N− N O (rather than 2 2N for the total space). Previously studied QA approaches have relied on a transverse field driver Hamiltonians to solve CFD problems [4, 32, 33]. As noted 62 (1) (2) (3) (4) Figure4.6: Logarithmofsuccessprobabilityasafunctionofinstancedegeneracyandnumber of faults in the MFD (for different values of the total annealing time T f ). 63 in Ref. [33], more novel driver Hamiltonians are one of the clearest routes to improving the performance for QA on such classes of problems and this chapter can be considered a step in this direction, relying specifically on the development and theory of CQA [19, 20, 34]. Ordinary transverse field quantum annealing approaches typically require imposing penalty terms to suppress configurations that do not satisfy the problem constraints. Our approach exploits the structure of the problem to construct a set of special drivers (whose number is linear on the size of the circuit) that naturally implement transitions only between valid configurations. We constructed a family of circuit instances generalizing the ISCAS C17 (17 wires, 5 inputs, 2 outputs) and analyzed the performance of the delineated protocol on these instances. By analyzing the spectral gap of a set of random instances, we were able to exploit its features (mainly the approximate location of the minimum gap) to construct an annealing schedule that evolved faster in regions of large spectral gap. This information was then utilized to benchmark the performance of the approach on a family of larger circuits with26wires,6inputs,and4outputs(whichwerefertoasC26). Welookedattheminimum gap for instances with non-degenerate solutions (i.e., those having unique solutions with the minimum number of faults), and found that the size of this gap had a mild dependence with the number of faults in the MFD. Thisisofinterest, becauselargecircuit(∼ 10 6 wires)CFDproblemstypicallyfoundin industrial applications are currently solved using a brute force search approach, by looking for diagnosis with a fixed number of faults (typically 1 or 2). Since the complexity of this approachscalesexponentiallywiththenumberoffaults,itiscurrentlynotpracticaltosearch fordiagnoseswithhighernumberoffaults. Incontrast,themilddependenceoftheminimum gap on the number of faults of the MFD suggests that our QA based approach could solve some instances with more faults in their MFDs, using a fixed amount of computational resources, since QA is essentially characterized by the minimum gap. 64 In our QA approach based on special drivers, we chose the sign of each term in a way to make the overall Hamiltonian stoquastic. We considered one possible way of making it non-stoquastic by changing the sign of each driver term. Since non-stoquastic Hamilto- nians could be more difficult to simulate, it has been speculated that they could provide more computational power. In our particular case, computing the spectral gap showed that the minimum gap was typically smaller than for the stoquastic case, which would translate into a lower success probability for solving our CFD problem. However, it is important to point out that there are many ways of setting the phases of the driver terms in order to make the Hamiltonian non-stoquastic. A different choice (maybe informed by the circuit structure) could induce more interference effects between different diagnoses during the evo- lution, thereby possibly enhancing those with a lower number of faults. This is an intriguing possibility that we plan to pursue in future work. In analyzing the performance of our approach, we focused on mainly two aspects we believe could prove beneficial when applied to larger scale problems. One was exploiting the generic form of the minimum gap as a function of the annealing parameter. We showed that it starts large, decreases almost monotonically and reaches a minimum in the last third of the anneal. These generic features allowed for a tailored schedule that moves faster at the beginning and slows down at the end where the minimum gap is typically located. We showed that this improves the probability of success for a given total anneal time, which will reduce the time needed to see a solution (i.e., the number of times the annealing should be repeated). Theotheraspectwestudiedwasthedependenceoftheminimumgaponthenumberof faults in the MFD (minimum fault diagnosis). Our simulations showed that the dependence is rather mild, and furthermore, the increased degeneracy seen in instances with a higher number of faults in their MFD translated to a higher probability of success for instances with larger number of faults in their MFD. This is of interest because in practice (i.e., in actual industrial applications), large instances of CFD are tackled using exhaustive search 65 for diagnoses with low number of faults (i.e., a few). Since the complexity of such search increases exponentially with the number of faults, this can become impractical for circuits with 10 6 , wires and diagnoses with more than 2 faults. Our analyses suggest that a QA approach could push this boundary further, resulting in significant practical (economical) impact. We would like to address the lack of head-to-head comparisons of our approach with classical algorithms. Our analysis involved simulating the quantum evolution of systems that required considerable computational effort (C26 was represented using 22 qubits in the reduced subspace). However, these circuits are still small for classical methods: since the numberoffaultsisupperboundedbythenumberofcircuitsoutputs,ourexampleswerestill intheregimewhereexhaustivesearchwouldbeextremelyfastandclassicalalgorithmsmore closely related to QA, such as simulated annealing, can be misleading to make comparisons with. Hence, we considered that a head-to-head comparison of time to solution for example, would not be very illuminating. Instead, as discussed in the previous two paragraphs, we decided to focus our analysis on features of the QA approach that we believe could provide an improvement when applied to larger circuits, where the combinatorial hardness of CFD becomes unavoidable for classical algorithms. Finally,wewouldliketonotethatourapproach,eventhoughdesignedwithaquantum annealingframeworkinmind, canbetranslatedtothecircuitmodelofquantumcomputing, followingtheframeworkoftheQAOAapproach. Forexample,theconstructionofindividual drivertermsaroundgatescanbetranslatedintounitarymixingtermsfortheansatzstrategy based on the same underlying structure. Furthermore, since the structure of these driver terms is related to the circuit structure, it could be possible to gain some insight into better waysofchoosingthemixingparameters,whichcouldimprovetheperformanceoftheQAOA approach. We consider this an intriguing direction for future research. 66 4.6 Acknowledgements for Chapter We thank Sandeep Gupta for many useful conversations about the CFD problem. The research is based upon work (partially) supported by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA) and the Defense Advanced Research Projects Agency (DARPA), via the U.S. Army Research Office contract W911NF-17-C-0050. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the ODNI, IARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon. 67 Chapter 5 Tailored Quantum Alternating Operator Ans¨ atze for Circuit Fault Diagnostics Similar to Chapter 4, we consider in detail the CFD problem, but from the stand point of theQuantum Alternating Operator Ansatz. Inthis Chapter, weconstruct severalans¨ atze tailored to solve the combinational circuit fault diagnostic (CCFD) problem in different subspaces related to the structure of the problem, including superpolynomially smaller sub- spaces than the whole Hilbert space. We introduce a family of dense and highly connected circuits that include small instances but can be scaled to larger sizes as a useful collection of circuits for comparing different quantum algorithms. We compare the different ans¨ atze on instances randomly generated from this family under different parameter selection methods. The results support that ans¨ atze more closely tailored to exploiting the structure of the underlying optimization problems have better performance than more generic ans¨ atze. This chapter is based on work in Ref. [84]. 5.1 Introduction We introduce and characterize several different Quantum Alternating Operator Ansatz (QAOA) [29] approaches to solving the Combinational Circuit Fault Diagnostic (CCFD) 68 problem, a combinatorial optimization problem of importance to diagnosing faults in cir- cuits. In particular, we identify different constrained spaces of interest and explore various constructions of mixing and cost operators that allow us to evolve within each constrained subspace,allowingtheprotocoltofocusonbringingthesupportofthesystem’swavefunction to an optimal or approximately optimal state within this subspace that can be significantly smallerthanthespaceofphysicalqubitsneededtorunthesystem. Therearemanysuchrel- evant subspaces and through a different selection of mixing and phase-separation operators even the same subspaces can be explored in very different ways. After describing QAOA for optimization problems, we delineate the stuck-at-fault model of CCFD considered in this chapter and how can be cast as an optimization prob- lem. We then develop several different ans¨ atze for solving this optimization problem, be- ginning with the least constrained (measured as the resulting size of the constrained space maintained) and ending with the most constrained. The most constrained has a superpoly- nomially smaller constrained space. For circuits with a logarithmic or less minimum fault explanation, it can be modified to have a constrained space that grows subexponentially in the number of wires while the most generic approach grows exponentially. Methods used for constructing ans¨ atze in this domain can be useful for construct- ing more tailored approaches for other domains, which is an interesting avenue for future quantum algorithms. Results from running each ansatz under several parameter selections indicate that ans¨ atze more closely tailored to a problem can have better performance than more generic ans¨ atze. The relative success of simple parameter selection methods used in this chapter are of interest to researchers focusing on QAOA for other domains. For many classes of combinatorial optimization problems that are NP-hard, the fea- tures of specific instances can make them in practice more or less accessible to different algorithms. The features of typical CCFD instances, such as a planar-like topology that is often amendable to embedding on local two dimensional lattice quantum architectures, makes it a strong candidate for quantum algorithms utilized on NISQ devices. Given the 69 real-life applications of CCFD, it has become a problem of interest for leveraging potential quantum advantage for industrial purposes [4]. Our results in this chapter support that tai- lored ans¨ atze should be explored on NISQ quantum computers to solve the CCFD problem. The family of scalable, dense, and highly connect circuits we introduce are useful testbeds for such devices. Even between different QAOA algorithms, the features of particular instances can affect which approaches are more or less promising. For example, we consider an ansatz that is similar to a recently studied CQA approach [23] that shows promise in the regime where an instance has higher degeneracy in the solution space and higher minimum fault explanations while another approach maintains a subexponential constrained space when there is an upper bound on the minimum fault explanation; this suggests the scaling of the minimum fault explanation for instances of interest is important to determining the best ansatz to utilize. However, there are still more considerations for implementing these approaches on quantum systems, including the preparation of the initial state and considering the best implementation of the required unitaries through a specific gate set. Certainly more than the size of the constrained is important for determining the best ansatz for a problem and it remains an open challenge to identify what specific features of mixing operators and phase separating operators would lead to the best performance. In Section 5.2, we introduce QAOA for combinatorial optimization problems and de- velop the principles around constrained quantum evolution within this context. In Section 5.3, we describe the single pair input-output circuit fault diagnostic problem over the stuck- at fault (SAF) model, where circuit wires are either healthy or permanently stuck at one or zero. In Section 5.4, we delineate different QAOA protocols, with different resulting con- strained spaces. In Section 5.5, we introduce a family of circuit instances to explore the suitability of our QAOA approaches to solve this problem for small sizes. In Section 5.6, we benchmark the approaches on a distribution of small sized circuit instances introduced in 70 Section 5.5. Our results support that tailored ans¨ atze can be beneficial to obtaining better performance on optimization problems like CCFD where there is structure in the underlying problem that can be exploited. 5.2 ConstrainedEvolutioninQuantumAlternatingOperator Ans¨ atze For combinatorial optimization problems, there have been recent advances in the re- search of applying quantum algorithms to solve such problems that have centered around the Quantum Approximate Optimization Algorithm (QAOA1) [24], which is a very general protocol to approximately (or exactly) solve combinatorial optimization problems, such as the Max Cut problem [24, 25, 70, 85]. However, further generalizations of this concept, such as RQAOA [86] and Quantum Alternating Operator Ansatz (QAOA) [27, 29, 85], have been developed in the hope to better tailor protocols to particular problems and thereby exploit their specific structure. For the purpose of this chapter, we focus on using QAOA protocols to find optimal solutions to a combinatorial optimization problem. For example, Ref. [85] demonstrates a near-optimal quantum unstructured search algorithm based on using coher- entphase-separationoperatorsintheQAOAsetting. Underreasonablecomplexity-theoretic assumptions, QAOA cannot be efficiently simulated by any classical computer [87]. For a small number of qubits, it has been realized on a NISQ [5] device [88]. GivenageneralbinaryoptimizationproblemwithN bits,thequantumcomputerworks with N qubits, over a space C 2 N . The wavefunction of the system, |ψ ⟩, is a normalized complex-valued vector in this space and potential actions on this state (i.e., algorithmic steps) are represented by unitary operators. QAOA works by starting the quantum system in a prepared state and then applying a seriesofangle-parameterizedunitariestoevolvethesystemintoastatethatisthenmeasured over a predetermined basis. The result can then be interpreted as a bit-string solution to 71 the optimization problem. QAOA divides the task of evolving the system into P rounds, each involving the application of two sets of operators. The first is a set of mixing operators, discussedingreaterdetailslater. Thesecondisadiagonalunitaryoperatorassociatedwitha classical costfunction that actsto essentiallyevaluate thequality ofdifferent configurations. Given a classical cost function C(x), in both QAOA and QAOA1, Hamiltonian form phase-separation operators are used: U c (α ) = P x∈{0,1} N e iαC (x) |x⟩⟨x|, where |x⟩ are the orthonormalcomputationalbasisstatesin C 2 N . Anysuchstate|x⟩(aket-state)canbewrit- ten as an outerproduct state of individual qubit states |x⟩ =|x 1 ,...,x N ⟩ =|x 1 ⟩...|x N ⟩ with x∈{0,1} N , |x i ⟩∈ C 2 , and x i ∈{0,1}. As such, the classical cost function is inter- pretedasaclassicalHamiltonianH c = P x∈{0,1} n C(x)|x⟩⟨x|(areal-valueddiagonalmatrix in C 2 N ). The complex exponentiation of a Hermitian operator is a unitary operator and so U c (α ) is a diagonal unitary operator. For example, a marked state cost function associated with a single solution state x ∗ = (1,...,1) could be C(x) = − Q n i=1 x i . Then U c (α ) = 1− (1− e − iα )|x ∗ ⟩⟨x ∗ |. The wavefunction associated with the quantum system, |ψ ⟩ = P x∈{0,1} n a x |x⟩, will therefore evolve to |ψ ⟩ = a x ∗ e − iα |x ∗ ⟩ + P x∈{0,1} n /{x ∗ } a x |x⟩ and so the behavior of U c (α ) is to add the same phases to computational basis states that have the same energy evaluations according to the cost function C(x). QAOA1 is a type of QAOA approach with a specific mixing operator and a fixed startingstate. BothoftheseconditionsarereplacedwithamoregeneralconditioninQAOA. In QAOA1, we begin in the uniform superposition state|+⟩ ⊗ n = 1 √ 2 n P x∈{0,1} n |x⟩, so that every computational basis state |x⟩ has the same amplitude. The mixing operators are one-local Pauli X operators: U d (β )= n Y j=1 e − iβσ x j . 72 The action of σ x j on qubit j is to flip the bit, mapping |1⟩ to |0⟩ and vice versa. Any QAOA approach that utilizes U d mentioned above is considered a QAOA1 approach for the purpose of our descriptions. As such, QAOA1 proposes a very specific type of ansatz, in which the mixing operator takes this form for any problem. QAOA will generalize this to allow for more structured ansa ¨tze. Then for a single round, t i of QAOA (including QAOA1), we pick two degrees α i ,β i and evolve the wavefunction as |ψ (t i )⟩ = U d (β i )U c (α i )|ψ (t i− 1 )⟩. U c changes the phase of states based on their evaluation from the classical cost function in the computational basis while U d acts as a mixer, leading to interference between the states in the superposition and potentially leading to concentration on low lying energy states based on the cost function C(x) after several rounds. Indeed, as the number of rounds P goes to infinity, one can select α i ,β i such that|ψ (t P )⟩ is guaranteed to minimize the cost function C(x) [24]. For a given finite P, there are 2P angles to select for running the algorithm and therefore can be cast as an optimization task where one wishes to find the angles that lead to a minimization of the cost function C(x). QAOA1, as well as QAOA, are a type of VariationalQuantumAlgorithm(VQA)[6]forwhichfindingoptimumparametersistypically an NP-hard problem [89]. The landscape of the final cost given by the quantum algorithm is known to suffer from a barren plateau [6, 90–92] and optimization with random starts can leadtoconvergencetolocalminima[6,25]. Nonetheless,optimumornearoptimumsolutions can follow similar patterns across instances [25, 26]. For a more detailed discussion, we refer the reader to Ref. [6]. Techniques utilized in this chapter will be discussed Sec. 5.6. Once the algorithm has run for P rounds, the wavefunction of the system is measured in the computational basis {|x⟩|x ∈ {0,1} n } such that the wavefunction collapses to a single state in that basis according to the Born rule: Pr(x)=|⟨x|ψ (t P )⟩| 2 . Formanyclassesofproblems, includingcertainclassoflinearandquadraticconstraint problems, it is possible to find alternative mixing operators that allow us to limit the evolu- tion of the wave function to a feasible subspace of a collection of those constraints [19, 20, 73 29, 34, 69]. The general problem is NP-Hard, but there is a simple polynomial algorithm for bounded operators [34]. In QAOA1, the mixing operator utilized can be associated with there being no constraint placed on the evolution, since every state is reachable under its action (although it is clearly not the only such mixing operator). The two essential re- quirements for the mixing operators are that they must take actions moving states in the constrained space to potentially any other state in that constrained space but cannot move such states outside of the constrained space. As such, QAOA describes a more general approach than QAOA1 in which the mixing operator and phase-separating operators can be much more general. In particular, they are usually tailored to the specific type of symmetries of the underlying problem (as the ansatz to the algorithm). Notice that QAOA shares the same acronym as QAOA1 in the literature. For example, Ref.[29] lists a compendium of mixing and phase-separating operators that can be utilized for different combinatorial optimization problems. Specific type of mixers have been studied more extensively; for example, XY-mixers are associated with a very simple kind of equality constraint that makes them useful for many types of combinatorial optimization problems [31] as well as quantum chemistry [93]. 5.3 Circuit Fault Diagnostics Diagnosing errors and faults for gate based digital circuits is an area of intense research as large scale integrated circuits and specialized circuit designs have become abundant in manyscientificandengineeringdisciplines. Increasinglysophisticatedautomationtechniques used to check and correct errors in the circuit design and fabrication are increasingly relied upon in practice. Because of the underlying combinatorial nature of this problem as well as the local design of many modern circuits, this is a class of problems that is well suited for typical NISQ [4, 5] architectures. 74 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Figure 5.1: All possible valid fault configurations for a small circuit with one FAN gate and one NAND gate that has a faulty output. The diagrams on the top left and right show minimum fault explanations for this instance. We employ a simple stuck-at fault model for analyzing circuits with a given string of inputs and empirically found string of outputs [72]. Let n be the number of wires in the circuit and n o be the number of output wires. Each wire in the circuit is either healthy or stuck at either zero (SA0) or one (SA1). Under the assumption that faults are equally likely to occur on every wire, the task of Circuit Fault Diagnostics (CFD) is an optimization problem to find the minimum number of faults needed to explain the input/output pair [4]. For the purpose of our discussion in this chapter, we consider gates that are one input/one output, two input/one output, or one input/two output. One input/one output gates are an identity gate ID, or an inverter gate INV. Two input/one output gates are OR, AND, XOR, NOR, NAND, or XNR. One input/two output gates are a fan out gate FAN, a fan out gate with an inverter on the first output F10, a fan out gate with an inverter on the second output F01, or a fan out gate with an inverter on both F11. 75 A configuration for the problem is a 2 n-bit string, where over bits 1,...,n are the wire bitsandoverbitsn+1,...,2narethefaultbits. Avalidconfiguration( w 1 ,...,w n ,f 1 ,...,f n ) has an erroneous value of a wire w i iff the corresponding f i is nonzero. The inputs to a circuit are considered to also be potentially faulty, such that input wires themselves could be SA0 or SA1, but output wires have to correspond to the empirical value they have (otherwise this diagnosis would be invalid), but that value may come from error propagation in the circuit or from the output value itself being SA0 or SA1. For whatever faults are present in the circuit, the output values can always be made to match the empirical value by taking the nonconforming output values and flipping their value (as well as their associated fault bit). As such the size of the valid configuration space is 2 n− no [23], since we have precisely n− n o wire locations where the fault flag can be 0 or 1 (the fault flag on the n o output wires is then forced to be such that the configuration is indeed valid). Fig. 5.1 shows all possible valid fault explanations for a small circuit. Note that there are 3 non-output wires and 2 3 valid fault configurations. 5.3.1 Valid and Invalid Configurations Around Gates Enforcing the logic of the circuit on our configurations is essentially enforcing the logic of each individual gate on the part of the configuration that correspond to that gate’s inputs and outputs. Valid configurations around gates follow the same logical delineation as valid configu- rations for the entire circuit. For any circuit or subcircuit, including a single gate, we can list the potential valid configurations by considering the different input/output pairs. If a configuration is valid for a particular input/output pair, it cannot be valid for any other and so the valid configurations always make disjoint sets. For that circuit or subcircuit, as discussed before, the inputs can potentially be faulty, but the outputs must match their empiricalvalue. However,thesizeofthedifferentvalidconfigurationsetsisalwaysthesame. 76 As such, the valid configurations around a gate are split by what the inputs before considering the fault status are (if they differ from this value, there must be an associated fault on this input) and what the output after considering the fault status is (i.e. the same situation as for the circuit at large). In Tab. 5.1 gives a full description of all the valid subconfiguration spaces for a NAND gate, where the first element in each entry is the trivial valid configuration that exists for every subcircuit (or the whole circuit) by applying the faults on the outputs only (i.e. all wires are healthy, except the outputs that disagree with the logical value they should have). If a configuration is valid and maintains a state status as described by an entry in Tab. 5.1, then swapping this subconfiguration with a state status in the same entry will also bevalid, sincewedidnotchangethevaluethatcomefrompreviousgatesoraresetasglobal inputs to the circuit nor do we change the value of the output (which is then used as an input to a gate or is a global output for the circuit). For a circuit with a collection of gates G, we consider the input and output wires (I g ,O g ) for that gate. The collection of valid configuration sets around this gate, V g ={V (S Ig ;S Og ) g |S Ig ∈{0,1} |Ig| ,S Og ∈{0,1} |Og| }, (5.1) are indexed by what the potential values the wire inputs, S Ig , are set to without faults and what the potential wire outputs, S Og , are set to. For example, S Ig for a NAND gate in the fault model is a collection of the possible inputs{(0,0),(0,1),(1,0),(1,1)} and S Og is a collection of the possible outputs{(0),(1)}. 5.4 QAOA approaches to CFD In this section, we give details of the construction for several QAOA approaches to solving the CFD problem, with each constraining the evolution of the wavefunction to a 77 entry IO pair valid configurations 1 (0,0;0) (0,0,0;0,0,1), (1,1,0;1,1,0), (1,0,0;1,0,1), (0,1,0;0,1,1) 2 (1,0;0) (1,0,0;0,0,1), (1,1,0;0,1,0), (0,0,0;1,0,1), (0,1,0;1,1,1) 3 (0,1;0) (0,1,0;0,0,1), (1,1,0;1,0,0), (0,0,0;0,1,1), (1,0,0;1,1,1) 4 (1,1;0) (1,1,0;0,0,0), (0,0,0;1,1,1), (1,0,0;0,1,1), (0,1,0;1,0,1) 5 (0,0;1) (0,0,1;0,0,0), (1,1,1;1,1,1), (1,0,1;1,0,0), (0,1,1;0,1,0) 6 (1,0;1) (1,0,1;0,0,0), (1,1,1;0,1,1), (0,0,1;1,0,0), (0,1,1;1,1,0) 7 (0,1;1) (0,1,1;0,0,0), (1,1,1;1,0,1), (0,0,1;0,1,0), (1,0,1;1,1,0) 8 (1,1;1) (1,1,1;0,0,1), (0,0,1;1,1,0), (1,0,1;0,1,0), (0,1,1;1,0,0) Table 5.1: Each kind of input/output pair to a faulty NAND is described. Each entry has differentvalidconfigurationsandconfigurationsinthesameentryformasubspacesuchthat the action of the driver terms we construct connect the configurations of this subspace. specific subspace and using mixing terms which connect all feasible states within this space to one another in different ways. We map each bit in the 2 n-bit string to a qubit such that thewavefunctionof oursystem|ψ ⟩ isanormalizedvectorin C 2 2n . AttheendoftheQAOA algorithm, we measure|ψ ⟩ in the computational basis to extract a solution to our problem. 5.4.1 Approach 1: Transverse Field with QUBO This approach is closely related to a Hamiltonian description of this problem, similar for example, to the approach described in Ref. [4] for using a transverse field and an Ising Hamiltonian wtih explicit fault mappings to represent this problem. As such, our approach is in the spirit of QAOA1 for Ising spin problems. The mixing operators are the Pauli X operators: U d (β t )= n− no Y i=1 e − iβ tσ x i . 78 Since the output values of the circuit have to match the given empirical values, they are not allowed to change during the problem. As such, these qubits can be integrated out of the problem. The phase separating operators are associated with an Ising Hamiltonian: U c (α t )=e iα tH f , where the Hamiltonian counts the number of faults and enforces the logic of the gates: H f = 2n− no X i=n− no+1 (1− σ z i )/2 ! +κ X g∈G H g ! , whereH g = P V r g ∈Vg 1− P v∈V r g |v⟩⟨v| associatesacosttothestatebeinginaninvalid set for that particular gate. If a configuration x fails to be a valid configuration in the fault around a gate g, then H g will associate a cost to the state x. To ensure consistency with the model, we can set κ = n o +1, the number of outputs, so that it is never advantageous to break a gate over placing the faults at the very end of the circuit. The number of output bits is an upper bound on the number of faults of the minimal fault diagnosis; for individual instances, we can set κ to any value greater than the number of faulty output bits, which is less or equal to n o . As in QAOA1, our initial state is the uniform superposition over 2n− n o qubits: |ψ ⟩=|+⟩ ⊗ 2n− no = 1 √ 2 n− no P x∈{0,1} 2n− no |x⟩. 5.4.2 Approach 2: Transverse and XY-mixer with QUBO Any solution to the single input/output CFD problem has at most n o faults. To exploit this bound, we can employ the XY-mixer, such that over the fault bits (f 1 ,...,f n ), we impose the constraint P n i=1 f i ≤ n o . This can be achieved in different ways, for example, by introducing n o ancilla qubits and employing the standard ring XY-mixer over the n fault bits and n o ancilla bits. 79 NotethattheXYmixeroperatingonacomputationalbasisstateofN qubitskeepsthe number of ones fixed. So if the initial wavefunction only has support on computational basis states with k ones (|ψ ⟩= P x∈{0,1} N s.t.|x|=k a x |x⟩), then the wavefunction after applying any collection of XY mixers will still have support only on computational basis states with k ones. To achieve states with the number of ones between 0 and k, we can add k ancillas and then apply the XY mixer on N +k qubits, such that when those mixers are applied to a wavefunction ψ N+k = P x∈{0,1} N s.t.|x|≤ k P y∈{0,1} k s.t.|y|=k−| x| a x+y |x⟩|y⟩ the constraint on the number of ones in the wavefunction is maintained. Then at the time of measurement, we simply discard the ancilla qubits and consider the resulting state over N qubits. Since the outputs, like in the previous approach, can be integrated out and so the wire bits correspond to qubits between 1 and n− n o . The fault bits correspond to qubits between n− n o and 2n− n o and the ancillas to qubits between 2n− n o +1 and 2n. Utilizing the XY mixer over qubits between 2n− n o +1 and 2n and the transverse field applied on the wire bits leads to: H d = n− no X i=1 σ x i ! + σ x n σ x n− no+1 +σ y n σ y n− no+1 + 2n− 1 X i=n− no+1 σ x i σ x i+1 +σ y i σ y i+1 , which can be split into two noncommuting Hamiltonians: H 1 = n− no X i=1 σ x i ! + σ x n σ x n− no+1 +σ y n σ y n− no+1 + ⌊(n+no)/2⌋ X i=1 σ x n+2i σ x n+2i+1 +σ y n+2i σ y n+2i+1 , and H 2 = ⌊(n+no)/2⌋ X i=1 σ x n+2i− 1 σ x n+2i +σ y n+2i− 1 σ y n+2i . 80 Whenn+n o is odd, H 1 also has a noncommuting term associated with qubits n− n o +1, n− n o +2, and2n. AsXYmixingoperators[31], wecanapplycommutingmixingoperators associated with H 1 and then H 2 as: U d (β t )= n− no Y i=1 e − iβ tσ x i ! ⌊(n+no)/2⌋ Y i=1 e − iβ t(σ x n+2i σ x n+2i+1 +σ y n+2i σ y n+2i+1 ) e − iβ t(σ x n σ x n− no+1 +σ y n σ y n− no+1 ) ⌊(n+no)/2⌋ Y i=1 e − iβ t(σ x n+2i− 1 σ x n+2i +σ y n+2i− 1 σ y n+2i ) . Thephase-separatingoperatorsarethenthesameasintheprevioussection. Theinitial state is an outer product (concatenation) of the uniform superposition over wire bits and a statewithauniformsuperpositionovern+n o stateswithpreciselyn o bitssettoone: |ψ ⟩= |ψ 1 ⟩|ψ 2 ⟩with|ψ 1 ⟩= 1 √ 2 n− no P x∈{0,1} n− no |x⟩and|ψ 2 ⟩= n+no no − 1/2 P x∈{0,1} n s.t.|x| 1 =no |x⟩. Wediscussthepreparationofthewell-knownentangledstate|ψ 2 ⟩studiedinseveralcontexts with relation to the XY mixer in Sec. 5.4.6. 5.4.3 Approach 3: Graph Diffusors with Linear Field on Fault Bits In this approach, we tailor the mixing operators to maintain the valid fault configu- ration space. Around each gate, a mixing operator is associated with swapping valid fault configurationsaroundthatgate,utilizingthedescriptionsgiveninSec.5.3.1suchthat,given avalidconfiguration, themixingoperatorspopulatestatesthatarealsovalidconfigurations. Given a set of logical gates G ={g 1 ,...,g m } in the fault model over w 1 ,...,w n wires and f 1 ,...,f n fault bits, define V (S I ;S O ) g as detailed in Sec. 5.3.1. We wish to construct the collection of unitariesU ={U (1) g 1 ,...,U (r) g 1 ,...,U (r) gm } associated with the mixing operator of each gate and the respective tuple of input/output pairs for each gate (note that the index (r) runs over a set of input/output to the gate, so it is typically an n-tuple, and not a single integer). For example, as in the circuit considered in Sec. 5.3 and illustrated in Fig. 5.1, if 81 g 2 is a NAND gate then {U (1) g 2 ,...,U (8) g 2 } are associated with the eight input/output pairs for a NAND gate. Define the uniform superposition over the collections V (j) g i as V (j) g i E = 1 q |V (j) g i | P v∈V (j) g i |v⟩ (see Tab. 5.1 for such a description for a NAND gate). Then define each mixing operator associated with a specific input/output pair for a specific gate: U (j) g i (β t )= 1− 1− e − iβ t V (j) g i V (j) g i = 1− 1− e − iβ t /|V (j) g i | X v,u∈V (j) g i |v⟩⟨u|. Then define the mixing operator associated with each gate: U g i (β t )= (r) Y j=(1) U (j) g i (β t ). U g i (β t ) are unitary because U (j) g i (β t ) are mutually commutative over input/output pair index j. U (j) g i has eigenvalue e ia for an eigenvector V (j) g i E and eigenvalue 1 for any vector in the range of the projector 1− V (j) g i ED V (j) g i . Then define: U d (β t )= m Y i=1 U g i (β t ), as the mixing operator for this ansatz, where we apply each gate one after another. However, for circuits with a highly regular structure (such as those introduced in Sec. 5.3), we can group many commutating unitaries together and apply all U i (β t ) with in two steps. The cost function (also one-local) simply counts the number of faults of the configura- tions: U c (α t )= 2n Y i=n e iα t( 1− σ z i )/2 . 82 The feasibility space and how it is connected through the action of U d is more compli- cated for this approach than previous approaches such that it cannot be as easily described by a uniform superposition over the states. We begin with a known feasible configuration and apply U d (β t ) to explore the feasible configuration from there. A simple starting state used for this purpose is the state with only faults placed on the faulty output states. We also consider a modified protocol which includes a second cost function that has thisstateasitsminimum,inclearinspirationfromasimilarCQA[23]approach. Asufficient (and one-local) cost function puts a penalty on every bit that does not conform with the initial state chosen. As such the cost associated with any state in the space is given by its Hamming distance from the initial state. Let x 0 = {x 0 1 ,...,x 0 2n } be the computational basis string associated with the initial state|ψ ⟩=|x 0 ⟩. Then the initial state cost function operator can be implemented as: U s (γ t )= 2n Y i=1 e iγ t( 1− (1− 2x 0 i )σ z i )/2 . For this modified protocol, we have three sets of operators: the initial state cost func- tion operator, the mixing operators, and the phase-separating operator. As such, we gener- alizetheQAOAprotocoltohavethreeanglestoselectforeveryround,suchthestateevolves under(α 1 ,β 1 ,γ 1 ,α 2 ,β 2 ,γ 2 ,...,α p ,β p ,γ p ). Forstept i ,|ψ (t i )⟩=U s (γ i )U d (β i )U c (α i )|ψ (t i− 1 )⟩. In Sec. 5.6 we consider performance of both approaches for small circuit instances and dif- ferent parameter selection methods. Since U s and U c are both one-local operators, it is also straight forward to apply them together on a device. 5.4.4 Approach 4: Transverse Field on Faults with Oracle Circuit Simulator Rather than representing the wire and fault bits explicitly, in our next approach we focus on the space of valid fault configurations (of dimension 2 n− no ). Notice that for any 83 invalid fault configuration, it can be made a valid configuration by simply flipping the fault bit for the output bits which are incorrect. Consider a fault configuration f = (f 1 ,...,f n− no ) over non-output wires. The energy of the state is dependent on the number of nonzero fault bits P(f) = P n− no i=1 f i as well as the number of implied faults needed on the outputs to make this a valid configuration Q(f) = P n i=n− no (w i − S(f,i)) 2 , where S(f,i) is the value placed on output wire w i when simulatingthecircuitwiththegiveninputsandthefaultconfiguration f overthenon-output wires. Then R(f) = P(f) + Q(f) counts the total number of faults implied by the fault configuration f inthisconstrainedspace. Byconstructinganoraclethatsimulatesthecircuit and then uses this information to find the proper fault count, we have a phase-separation operator with the same cost function as in Sec. 5.4.3: U c (α t )= X x∈{0,1} n− no e iα tR(x) |x⟩⟨x|. As in QAOA1, our starting state will be|ψ ⟩=|+⟩ ⊗ n− no = 1 √ 2 n− no P f∈{0,1} n− no |f⟩ and we use the one-local Pauli X mixing operators: U d (β t )= n Y i=1 e − iβ t(σ x i ) . To see how U c (t) can be implemented in practice, we consider a protocol where ancilla qubits are used for the computation of the faults needed on the outputs. Let |ψ (r i− 1 )⟩ ∈ 84 C 2 n− no bethewavefunctionatthebeginningofroundr i ,withtheinitialwavefunctiondefined as above. We show how|ψ (r i )⟩ is then generated over the current round. For r i ∈[r 1 ,r t ]: |ν (r i )⟩=U circ |0⟩ n |ψ (r i− 1 )⟩|0⟩ no = n X x∈{0,1} n− no ⟨x|ψ ⟩|f(x)⟩ n |x⟩ n− no |g(x)⟩ no (Simulate Circuit) |µ (r i )⟩=U c (α i )|ν (r i )⟩ (Apply Phases) = 2n Y j=n+1 e iα i (1− σ z j )/2 X x∈{0,1} n |f(x)⟩) n |x⟩ n− no |g(x)⟩ no = X x∈{0,1} n− no 2n X j=n+1 e iα i x j ! |f(x)⟩ n |x⟩ n− no |g(x)⟩ no |0⟩ n |ϕ (r i )⟩|0⟩ no =U † circ |µ (r i )⟩ (Undo Simulation) |ψ (r i )⟩= n− no Y j=1 e − iβ i σ x j |ϕ (r i )⟩ (Apply Mixing) Here, U circ implements the simulation of the faulty circuit such that f(x) is the valid configuration over the wire bits that |x⟩ specifies from the non-output fault bits, while g(x) are the required faults on the output bits such that x⊕ g(x)⊕ f(x) is a valid configuration over the 2n bits. 5.4.5 Approach5: BoundedFaultCountwithOracleCircuitSimulator In this section, rather than restricting ourselves to the subspace of all fault configura- tions,wewishtorestrictourselvestofaultconfigurationsuptoamaximumnumberoffaults. To accomplish this task, we can use the XY-mixer also discussed in Sec. 5.4.2, by utilizing ancilla qubits in a complimentary way. Here we have n− n o non-output fault flag bits (same asSec.5.4.4)andn o ancillaqubitstoallowforrepresentationofallstateconfigurationswith faults less than or equal to n o . 85 Approach Mixing Operator Cost Function Size of constrained space Round Complexity 1 Pauli X Ising O(2 2n ) O(1) 2 XY-mixer and X Ising O(2 n− no (n+n o ) no ) O(1) 3 Gate-based Diffusors Pauli Z O(2 n− no ) O(1) 4 Pauli X Circuit Oracle O(2 n− no ) O(C p ) 5 XY-mixer Circuit Oracle O(n no ) O(C p ) Table 5.2: Depending on our selection of mixing and phase-separating operators, we can constrain the evolution of the wavefunction to a constrained spaces of differing sizes. The cost function remains the same as that used in Sec. 5.4.4 and the same procedure with the simulation of the circuit to find the faults on the end bits can be used, except the aforementioned replacement of the mixing operator. Theinitialstateisthen|ψ ⟩= n no − 1/2 P x∈{0,1} n− no s.t.|x| 1 ≤ no P y∈{0,1} no s.t.|y| 1 =no−| x| 1 |x⟩|y⟩, similar to the state mentioned in Sec. 5.4.2. Note that since we also have implied faults, the maximum number of faults that can be expressed is between n o and 2n o . Initializing this state is discussed in more detail in Sec. 5.4.6. 5.4.6 Size of the Relevant Constrained Space and Summarization for Approaches In each of the approaches delineated in Sec. 5.3, the size of constrained space of interest in which the evolution of the wavefunction is limited to differs. In Fig. 5.2, each approach is summarized by the type of mixing operator, the type of cost function, the size of the constrained space, and the complexity of each round. While the approach in Sec. 5.4.3 and Sec. 5.4.4 differ greatly in the type of mixing and phase-separation operators used, they constrain the system to the same subspace. Consider the exploration of this space under these two different approaches. The latter begins with a uniform superposition over all valid fault configurations and the neighborhood of each computational basis state under the action of each Pauli X operator in the mixer is a state 86 Unconstrained Space ~ Valid Faults ~ Valid Bounded Faults ~ Bounded Faults ~ Figure 5.2: A visual representation of the subspaces that the wavefunction is kept within during evolution through each QAOA approach considered in this chapter. that is a bit-flip away, like a high dimensional hypercube. The former begins with a single feasible state and the neighborhood of each state under the action of the mixing operator are dependent on available transformations around each gate to move it to a new fault configuration. Moreover, while the cost function associated with these two approaches are equivalent, the resulting phase-separation operators to implement this cost function are very different. For example, while each call to the cost function runs in O(1) for the fault counting in the former, each call to the cost function in the latter will require simulating the correspond- ing circuit, which has a particular depth C p . For example, for the distribution of circuits introduced in Section 5.5, the depth of the classical circuit grows as O( √ n) if we choose the diameter and depth to be equal (this relationship holds for the family considered for experiments in this chapter). 87 Approaches2and5requirethepreparationofaparticle-number-conservingstatestud- ied in many areas of quantum computation and quantum chemistry [94–97]. We refer the readertotheAppendixofRef.[31]foradiscussionofthisstateinthecontextofQAOAwith the XY mixer. In general, for a uniform superposition over n qubits preserving number k, the state can be constructed with n k =O n k CNOT gates [97], which can be prohibitive for a circuit with a large bound on the number of faults. However, Ref. [94] provides a projective measurement method that grows polynomially with n and independent of k for large n. Alternatively, one can use an approach similar to Approach 3 where the initial state is a computational basis state in the feasible space. Initial state selection for QAOA remains an important area of current research [31], including warm starts [30, 98]. Moreover, the number of minimum faults for many circuits of interest can be much less than n o and one could adapt the approaches to utilize this smaller space. 5.5 Random CFD instances with Balanced Width and Depth TodemonstratehowthesedifferentapproachespracticallyperformontheCFDproblem, weintroduceafamilyofrandominstancesthatscalewithasinglesizeparameterandallowfor detailedanalysisonsmallsizes. Fig.5.3showsthefirstfewsuchcircuitswithincreasingsize. The central feature is that both the depth of the circuit and the number of inputs/outputs (width) scale together with the size, number of wires, of the circuit. We randomly select inputs to the whole circuit and apply faults to the outputs such that every output is faulty. We randomly select the inputs and gates at each spot from the relevant gate set for every size to generate the instances used in Sec. 5.6. The valid fault configurations for an instance of the smallest circuit in the family are in Fig. 5.1. Unlike all the other circuits, this has only one output and so we modify the 88 Figure 5.3: Diagrams for circuit instances of the CFD problem considered in Sec. 5.6, de- pending on the number of qubits, from these small, local, and dense circuits. Every box is a two intput/one output gate, every dot is a one input/two output gate, and every diamond is a one intput/one output gate. If a top wire has a ⊤ and a bottom wire has a ⊥ at the same depth, they refer to the same wire (which has been wrapped around). model for this small circuit for our experiments such that placing a fault on the output wire requires two fault whereas every other location requires one. While on large scales we expect the performance of any specific ansatz strategy to decrease with the problem size, at the small sizes used for our simulations the performance can vary by problem size in a more complicated fashion. For all instances the maximum number of faults needed is two, since there are two outputs. We generate 100 instances that are filtered such that the minimum fault explanation requires a single fault. Approach 3, unlike all the other approaches, would be initialized in a solution state without this filtering. For larger sized circuits of this family the chance of a randomly se- lectedinstancehavingaminimumfaultexplanationthatsaturatesthisboundisincreasingly diminished, but this filtering is an important step for these small size circuits of this family. On the filtered data sets of different size, the specific topology of the circuits can still lead to important differences for fault configurations. For example, the circuit depicted at the bottom middle in Fig. 5.3 has much more diversity in the typical size (or degeneracy) of 89 the solution space compared to smaller sizes. Solution degeneracy has been seen to have a beneficial impact on a variety of optimization problems for QAOA. 90 5.6 PerformancewithDifferentParameterOptimization (a) (b) (c) (d) (e) (f) Figure 5.4: Performance of ans¨ atze with different parameter selections. The solid lines indi- cate the performance on the median of the instances, while the ribbons correspond to the lower and upper quantiles. Markers correspond to the success probabilities of individual instances. 91 Using the family of CFD instances from the previous section, we consider the per- formance of QAOA strategies with different methods of parameter optimization: BRUTE, INTERP, LINANGOPT, and LINCOEFOPT. Each of the first three requires the expected costofthefinalwavefunctiontoupdatetheangleparameters. Forimplementationonaquan- tum computer, the expected cost of the final wavefunction has to be sampled with repeated measurements and there will be trade-off between repeated runs to accurately compute this value and running the protocol with updated parameters based on the approximation [25]. Angles were selected on the interval [− π,π ]. For BRUTE, we choose 100 random seeds and run Nelder-Mead with 100 iterations to optimize each choice, using the final Hamiltonian as the cost function and selecting the choice that minimized the cost function the most. For INTERP, we use the interpolation functioncalledINTERPinRef.[25]thatshowedgoodperformancefromthismethodintheir results. We use Nelder-Mead (with 100 random seeds and 100 iterations each) on p=2 and then for p>2 begin with the linear interpolation from p− 1, optimizing with 100 iterations on this result. For LINANGOPT, we start the protocol with a linear ramp for each angle (i.e. for round k∈[1,p], α k =kπ/p and β k =π − kπ/p ) and then optimize the angles (with 200 iterations). For LINCOEFOPT, we use a coefficient Γ ∈{9/10,1/2,1/4,1/8,1/16} and use the same linear ramp, but replacing α k ,β k with α k Γ ,β k Γ and then optimize Γ over 20 iterations, selecting the best performin Γ. In each situation described Approach 3 (with an initial state cost function) is different fromtheotherapproachessinceithasthreeanglestoselectforasingleroundk. Astheinitial startingpointforLINANGOPTandLINCOEFOPT(withΓ), itbeginswith γ k =π − kπ/p , α k = Θ( k ≤ p)(2kπ/p ) + Θ( k > p)(π − 2kπ/p ) (reaching a maximum around p/2 and minimums at 0 and p), and β k = kπ/p . Note Approach 3 was also considered without an initial state cost function, results support that this seemed to benefit its performance. ResultsfromRef.[25]suggestthatNelder-Meadcanperformaswellasotheroptimiza- tion algorithms, such as the Broyden-Fletcher-Goldfarb-Shanno algorithm. Utilizing other 92 optimizers for these tailored ans¨ atze is an interesting area of future study [69, 99, 100]. We report results with p set to 2 and 3 for BRUTE, 5 for INTERP, 5 and 10 for LINANGOPT, and 50 for LINCOEFOPT. The relative simulation range capable on a workstation grows depending on the size of the relevant constrained space for the problem and so the more constrained approaches have an increased simulation range, as such we report results on larger instances for these approaches for each parameter selection. Fig. 5.4 details the performance of each ansatz with each parameter selection strategy. Results support that ans¨ atze which exploit a more constrained space by tailoring the operators used to the problem structure generally perform better on the CFD problem and therefore are important methods for improving the practical applicability of QAOA and other variational quantum algorithms. Indeed Approach 4 and 5 consistently performed the best and Approach 3 performed comparably with some parameter selections for certain sizes. Surprisingly though, Approach 1 performed quite strongly for LINCOEFOPT with p=50. Theperformanceacrossthesedifferentmetricsgenerallysupportsthatwhilealinear tramp function might be well suited for certain approaches, such as Approach 1 and 4, it is interesting to consider different functions to parameterized the angles for other approaches. Utilizing an initial state cost function showed advantageous for Approach 3 each parameter selection method, except LINANGOPT where the benefit was less clear. Performance showed high variance under BRUTE and INTERP for most approaches. Approach 3, in particular, seems to struggle more at very low p, perhaps because the initial state has full support on a particular solution and must use mixing operators to explore the space while the other approaches begin with a uniform superposition over all states in the particularsubspace. SinceINTERPutilizestheresultsfromBRUTEat p=2, thisdifficulty can translate to this approach as well. While INTERP can alleviate the high cost associated withbruteforceevaluationtofindgoodparameterswithstepwiseoptimizationas pisscaled higher, the parameters found at low p can potentially be associated with a local minimum 93 thatmaybethebestcostminimizationthealgorithmcanaccomplishatthisdepthandthen the algorithm is subsequently stuck around this choice at higher p. Approach 3 also shows striking performance differences between instance sizes, es- pecially for LINANGOPT with p = 5 and LINCOEFOPT with p = 50. Approach 3 is initialized to a computational basis state with faults on the outputs and so the distance, in terms of number of mixing operators, to find a minimum state as well as the degeneracy of the solution space may play a more acute role compared to other approaches. For example, as noted in Sec. 5.5, the instances with 13 wires have instances with higher degeneracy than those of lower sizes. This typical degeneracy growth with problem size for circuits of interest is one reason leveraging quantum algorithms may be beneficial. Note that results are based on the number of rounds, p, but the actual depth of a quantum circuit to implement each of the different approaches will differ, especially on larger circuit instances. Moreover, Approaches 2 and 5 used a starting state that requires more involved preparation before the procedure can begin. 5.7 Conclusion One of the most compelling areas for utilizing variational quantum algorithms on NISQ devices arises for solving optimization problems, including those with hard constraints. QuantumAlternatingOperatorAnsatz(QAOA)andConstrainedQuantumAnnealing(CQA) are methods that can enforce those constraints naturally throughout the protocol or anneal. Designing approaches that can usefully use the structure of different optimization problems remains an important task for developing state-of-the-art quantum algorithms. Designing mixing operators and phasing operators to a constrained space can be more general than satisfying a global hard constraint, such as satisfying many local constraints (such as the valid configurations around a faulty gate [23], Sec. 5.4.3), a practical bound on the maximum value a counting problem can yield (Sec. 5.4.2 and 5.4.5), or simply finding 94 a way to represent a Hamiltonian that is difficult to implement without ancillas (Sec. 5.4.4 and 5.4.5). Inthischapter,weconstructedseveralgeneralapproachestoexploitconstrainedquan- tum evolution to solve the CFD problem, some with superpolynomially smaller constraint subspaces. Whilesomeoftheseapproachesmayrelatedprotocolsthatcouldbeimplemented on a quantum annealer, others are more challenging and therefore better suited for QAOA or digital adiabatic simulation. We introduced a family of random circuits that are pa- rameterized by a single size parameter, which can be useful for benchmarking future NISQ devices. Simulation of QAOA protocols with optimized parameters, interpolated and op- timized parameters, and linear interpolation suggest that more advanced ans¨ atze can give better performance by utilizing the underlining structure of an optimization problem. As such, designing and experimenting with more novel operators for solving optimization prob- lems remains an important area for future research. Nonetheless, the initial states of several approaches are more involved to prepare and the unitaries necessitated for each ansatz re- quire further analysis under specific quantum architectures. There are many interesting areas of future work that arise from the constructions con- sidered in this chapter. It would be interesting to explore approaches such as Approach 5 with an initial state in the computational basis to relieve the cost to preparing a highly entangled initial state. Since the number of minimum faults needed for a circuit can be much smaller than the upper bound given, it would be of interest to explore several of the approaches where this is utilized to form a more constrained ansatz, especially Approach 5. For example, if the minimum number of faults needed is known to scale logarithmically, the constrained space can then scale subexponentially for Approach 5. Given the rich and grow- ing literature of approaches to find suitable parameters for QAOA, it would be interesting to utilize such approaches for the ans¨ atze introduced here, especially those that are less cost prohibitive for intermediate p values where the advantage of more tailored ans¨ atze could be 95 more pronounce compared to more generic ans¨ atze using the same parameter selection ap- proach. Given that near-term quantum devices are likely to be noisy and have inaccuracies in the application of gates, it would be interesting to compare these ans¨ atze in this regime and consider modifications. The performance of each ansatz vary over the instances and it would be interesting to analyze what kind of features of an instance can be predictive of the performance of an ansatz. Finally, it would be of interest to consider other optimization problems in which similar structures can be exploited to tailor ans¨ atze. 5.8 Acknowledgements for Chapter WearegratefulforthesupportfromtheNASAAmesResearchCenterandfromDARPA under IAA 8839, Annex 128. The research is based upon work (partially) supported by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA) and the Defense Advanced Research Projects Agency (DARPA), via the U.S. Army Research Office contract W911NF-17-C-0050. The author was also sup- ported by the USRA Feynman Quantum Academy and funded by the NAMS R&D Student Program under contract no. NNA16BD14C. 96 Chapter 6 Conclusion: Satisfying Classical Symmetries Can Guide Future Quantum Algorithm Design 6.1 Summary In this thesis we have delineated how to impose classical symmetries on quantum op- erators and shown how this task, when framed as appropriate decision problems, relates to several important complexity classes. Given that classical symmetries are usually exploited todesignstateoftheartclassicalalgorithms, thereisnoreasontothinkitwouldbeanydif- ferent for quantum algorithms. As such, it is likely that state of the art quantum algorithms of the future for optimization problems will have to be tuned to exploit as much symmetry as possible in their approach. We strongly believe the results contained within this thesis can aid researchers designing future quantum algorithms with this intention in mind. Our work is an important step to understanding how to impose such classical con- straint on quantum systems through the symmetries of the quantum operators governing their evolution. Chapter 2 gives an expression for linear constraints, which allows us to classify the complexity of many important related computational questions to this task and giveasimplealgorithmforsolvingitforlocalHamiltonians. InChapter3wegiveanexpres- sion for general constraints and give a simple sufficient condition for quadratic constraints. Together, these two chapters are useful for any researcher wishing to construct quantum 97 Equal Subset Sum CP-QCOMMUTE ILP-QCOMMUTE NP-Complete ILP-QCOMMUTE-NONTRIVIAL ILP-QIRREDUCIBLE-COMMUTE-GIVEN-k 0-1-LP-QCOMMUTE NP-Hard ILP-QCOMMUTE-k-LOCAL P 2-OR-MORE Subset Sum Figure 6.1: A Venn diagram of different complexity classes discussed in this thesis under the assumption that P̸=NP. 98 operators satisfying classical symmetries. In particular, we expect these results to resonate with quantum computer scientist developing quantum algorithms, which can use these re- sults to develop approaches with more tailored quantum operators or, conversely, recognize limitations of such constructions. For current NISQ devices, broadly falling into the categories of systems capable of doing quantum annealing in the continuous case or QAOA in the discrete gate-based case (among other VQAs), we have considered concrete constructions to exploit classical sym- metries to design quantum algorithms for the circuit fault diagnostic problem. Our results, as shown in Chapter 4 and Chapter 5, support that tailoring quantum algorithms to sat- isfy classical symmetry is advantageous over utilizing classical cost functions in the form of quadratic unconstrained binary optimization. A constrained quantum annealing or tailored QAOA protocol can significantly reduce the size of the relevant Hilbert space over which the search problem is solved, thereby setting the amplitude of irrelevant states to zero be- fore the search begins. In comparison, an unconstrained quantum annealing protocol or the original Quantum Approximate Optimization Algorithm will begin with a quantum uniform superposition of states, thus giving equal weight to feasible and unfeasible states alike. 6.2 Future Work The trade-off between selecting different mixing operators and phase-separating oper- ators is still not well understood for QAOA or is the trade-off for selecting different driver and cost Hamiltonians for CQA in most contexts. It is important to continue to explore, for example, the role that stoquastic Hamiltonians (or unitaries associated with stoquastic Hamiltonians) can play for CQA and QAOA. While knowledge of a feasible state within the space can lead to an intuitive approach for utilizing catalyst driver Hamiltonians or catalyst unitary operators, there could be other valuable and more efficient approaches to consider handling the initial state problem. The theoretical results of Chapter 3 could also be useful 99 in settings beyond classical optimization, for example in quantum chemistry and are worth exploringforalgorithmsinthiscontext. AcertainquestionleftopeninChapter3iswhether decidingifthereisaboundedweightoperatorwhichcommuteswithapolynomialconstraint or collection of such constraints is known to be in P or known to be NP-Hard. While CQA has primarily been consider in the context of adiabatic quantum evolution, it is worthwhile to consider the application of catalyst Hamiltonians with specific symmetries to increase favorable diabatic evolution at key moments of an anneal. 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Using models to improve optimizers for variational quantum algorithms. Quantum Science and Technology 5, 044008 (2020). 109 Appendices C 0-1-LP-QCOMMUTE is NP HARD Wereducethe0-1-LP-QCOMMUTEproblemtotheEQUAL SUBSET SUMproblem. Wedefine the EQUAL SUBSET SUM problem as before: Definition. EQUAL SUBSET SUM Given a set S = {s 1 ,s 2 ,...,s n }, with s i ∈ Z + , find two non-empty disjoint subsets, A,B such that P a i ∈A a i = P b i ∈B b i . TheEQUAL SUBSET SUMproblemisknowntobeNP-Complete[48]. Wemapaninstance of the EQUAL SUBSET SUM problem to the 0-1-LP-QCOMMUTE problem; the former of which is defined over a set S ={s 1 ,s 2 ,...,s n }, with s i ∈Z + . In order to connect EQUAL SUBSET SUM with solving a linear system over discrete variables (the key of Theorem 2.3.1), we will associate an assignment of integers in S to the two subsets A and B with a function u over S such that u = {u 1 ,...,u n } with u i ∈ {− 1,0,1}. We associate the value u i as the assignment u gives to integer s i . Slightly abusing notation, this defines a function on any subset M ={s m 1 ,s m 2 ,...,s m |M| } such that u(M)={u m 1 ,u m 2 ,...,u m |M| }. When discussing a subset of a single element s e , we also abuse notation to allow for u(s e ) = u e . We can then define an integer valued function E S (u) = P n s i ∈S u i s i . If we associate integers s i such that u i = 1 with integers in subset A, and those with u i = − 1 with integers in subset B, then we can rewrite it as E S (u) = P s i ∈A s i − P s i ∈B s i (note that u i = 0 means that the corresponding integer is not chosen for any of the two subsets). Then, EQUAL SUBSET SUM has a solution if and only if there is an assignment function u with a nontrivial image such that E S (u)= P n i u i s i =0. 110 However, we need a vector representation to exploit the structure of Theorem 2.3.1. Let s max be the maximum of S. We define S M as the matrix with the binary representations of S as its column vectors. Given s j ∈S, we define entry s M ij =s i j – referring to the i-th bit of integer s j . This defines a m× n matrix with m=⌈log(s max )⌉. The idea is that we wish to give each integer an associated binary vector such that multiplying a binary vector with S M corresponds to selecting that integer to participate in a sum. We refer to the vectorized form of u as ⃗ µ ∈{− 1,0,1} n such that ⃗ µ = (u 1 ,...,u n ). Since multiplying a matrix by a vector on the right results in a linear combination of the matrix columns, with the coefficients being the corresponding components of the vector, it would be tempting to assume that E S (u) = S M ⃗ µ , since the columns of S M are associated withtheintegersinS. ThenwewouldhavesomethinglikeE S (u)=0ifandonlyifS M ⃗ µ = ⃗ 0, providing our desired connection between Theorem 2.3.1 and EQUAL SUBSET SUM. Unfortunately this does not work, since the columns of S contain a binary representa- tions of the integers s i , while the expression E S (u) refers to the usual addition of integers, and not bit component wise addition. To illustrate what we mean with this, consider the (improper) set S ={1,1,2} which delineates: S M = s 1 s 2 s 3 1 1 0 0 0 1 Even though the associated EQUAL SUBSET SUM problem has a simple solution associ- ated with the function u={1,1,− 1} (assign the first two integers to subset A and the third to subset B), a simple calculation shows that S M ⃗ µ =(2,− 1)̸= ⃗ 0. From the example above we can see that what we are missing is a way of incorporating the “bit carry” that occurs in binary addition into the operations of regular matrix-vector multiplication. The main goal of this appendix is to show how this can be accomplished by embedding these matrix operations into a larger vector space. 111 In order to resolve this issue, we will introduce a mechanism to do generalized bit addition - bit addition that is generalized to when the bit values can both be positive and negative as well as zero. We add ancillary bitsA such that u ∗ is the assignment u expanded to this new space S∪A as u ∗ ={u 1 ,...,u n ,u n+1 ,...,u n+|A| }. Slightly abusing notation, for any subset M =M S ∪M A with M S ={s ms 1 ,...,s ms |M S | } and M A ={a ma 1 ,...,a ma |M A | }, we define u ∗ (M)={u ms 1 ,...,u ms |M S | ,a ma 1 ,...,a ma |M A | }. We construct new constraintsK such that E S (u) = 0 ⇔ E K (u ∗ ) = 0. Moreover, u ∗ will allow for a vectorized form ⃗ µ ∗ and K a matrix K M (see C.2) such that E K (u ∗ ) = 0⇔ K M ⃗ µ ∗ = 0. Intuitively, u ∗ picks coefficients forvaluesoverS andissubsequently forced totakevaluesonAcorrespondingtodoingvalid bit addition and only satisfies K if the bit entries of the total sum is indeed zero. Fig. 6.2 gives a visual description of the steps used to create our full reduction. As such, for a given set of integers S, we follow the reduction to construct a binary matrix K M such that the row vectors of K M define the constraint operators ˆ K i = P |S∪A| j=1 k M ij σ z j . This serves as the input binary LP to the oracle solver of 0-1-LP-QCOMMUTE to tell us if a Hamiltonian H exists such that H has an off-diagonal term in the spin-z basis that shows the existence of⃗ v, ⃗ w, which describe two subsets A and B as the solution to the given Equal Subset Sum problem. C.1 Generalized Full Adder In this section we describe how to build the basis for our reduction, which is to find a matrix such that the values u ∗ takes on the set S are added bitwise over the ancillary bits A. There will be specific ancillary bits such that the total sum that u ∗ takes on S can be deduced from its value on these bits. Consider again the simple example we introduced in the previous section. We will add ancillary variables such that their values are forced to be what is dictated by the bit addition of values in S. This can be summarized in Table 6.1. If u ∗ takes a particular value on two inputs a and b, then the table describes what value u ∗ 112 Figure6.2: Aflowcartdescribinghowourreductionworks,werecommendmotivatedreaders refer back to it as they read the reduction. An instance of EQUAL SUBSET SUM (box 1) is mapped into a binary constraint representation such that the sum function E defined over the assignment u is equivalent to sum(A)− sum(B) where u assigns variables to either A, B, or they are not used (box 2). To exploit Theorem 2.3.1, constraints C are mapped to constraintsK (box 3), such that assignment E K (u ∗ )=0⇔E S (u ∗ )=0. Unlike S,K allows for a simple matrix representation such that K M ⃗ µ ∗ = 0⇔ E K (u ∗ ) = 0 (box 4), where ⃗ µ is a naive vectorized form of u ∗ . Note that if u exists such that E S (u)=0, then many u ∗ exist such that E K (u ∗ ) = 0, but each reduces to the same u. The constraint version of K M can be embedded row-wise to define operators ˆ K 1 ,..., ˆ K S∪A as ˆ K i = P |S∪A| j=1 k M ij σ z j such that a 0-1-LP-QCOMMUTE oracle solves to show the existence of a driver Hamiltonian H d , which we can interpret back to see there must be a solution to EQUAL SUBSET SUM as well. 113 Inputs Output Primary Secondary a b s c s c -1 -1 0 -1 - - -1 0 -1 0 1 -1 -1 1 0 0 - - 0 -1 -1 0 1 -1 0 0 0 0 - - 0 1 1 0 -1 1 1 -1 0 0 - - 1 0 1 0 -1 1 1 1 0 1 - - Table 6.1: The generalized full adder; if u ∗ takes a particular value on inputs a and b, then u ∗ will be forced to take the corresponding sum (represented by s) and carry (represented by c) values. In the case that a+b is not a power of two, s and c have two possible values they can take. Here primary (secondary) operations correspond to the operations where the carry is set to zero (nonzero) if possible. will be forced to take on new ancillary values s and c (representing the sum and carry bits respectively). Like the ordinary adder, the generalized adder accepts all values such that u ∗ (a) + u ∗ (b) = 2u ∗ (c) + u ∗ (s) except now u ∗ (x) ∈ {− 1,0,1} for any x and so u ∗ (a) + u ∗ (b) ∈ {− 2,− 1,0,1,2}. Note that then the carry bit and the sum bit are not unique like in the case of the ordinary full adder. For example if u ∗ (a) = 1 and u ∗ (b) = 0, then it is possible that u ∗ (c) = 0 and u ∗ (s) = 1 like in the ordinary adder, but also that u ∗ (c) = 1 and u ∗ (s)=− 1. Since 2u ∗ (c)+u ∗ (s) is the same value for both, they are both technically valid. The operations keen to the ordinary full adder we refer to as primary and those that do not as secondary. When possible, a primary operation will set the carry bit to zero while a secondary operation will set the carry bit to either one or negative one. One may hope that we could force the primary mode of operation, but we could not construct a 0-1 matrix that could force these modes of operations over the secondary modes since our condition for satisfaction is through equivalence statements like u ∗ (a)+u ∗ (b) = 2u ∗ (c)+u ∗ (s), but no equivalence statement can state a preference in representation. While it does not affect the correctness of our result, it does mean that the number of solutions is not preserved in our 114 reduction - there are many valid u ∗ that reduced to a single u. The reduction is therefore not parsimonious. To enforce the generalized adder between two inputs and two outputs we need to gen- eratethecorrectsubmatrix. Giveninputs aandb, wedefinethematrixon a,b,s,c,x 1 ,x 2 ,x 3 - with x 1 ,x 2 ,x 3 being intermediating ancillas - as: GA M = a b x 1 x 2 x 3 c s a b 1 1 1 0 0 0 0 1 0 1 1 1 0 0 0 1 1 1 1 0 0 1 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 (C.1) As constraints, we can write it as: GA 1 (a,b,x 1 ,x 2 ,x 3 )=0, (C.2) GA 2 (x 1 ,x 3 ,s,c)=0, (C.3) GA 3 (x 2 ,x 3 ,s,c)=0, (C.4) GA 4 (x 1 ,c)=0, (C.5) GA 5 (x 2 ,c)=0, (C.6) GA 6 (x 3 ,s)=0. (C.7) For every generalized adder in Fig. 6.4 (as described in the protocol we gave in Section C.1), we have a submatrix over the corresponding variables. We give a simple case by case proof that GA M enforces u ∗ to be valid if and only if its entries satisfy 2u ∗ (c)+u ∗ (s)=u ∗ (a)+u ∗ (b) as seen in Fig. 6.1 in Appendix D. 115 C.2 The Simple Reduced Case Before we move on to give a general protocol for any given problem, we consider the simple case we described earlier with the integer (improper) set S = {1,1,2}. We give a slightly reduced description for this problem to show what the reductions typically look like. We implement a generalized adder for the bits s 1 1 and s 1 2 - introducing the ancillary bits k 1 1 ,z 1 1 that are the corresponding carry and sum bit. We then implement a generalized adder for the bits s 2 3 and k 1 1 - introducing the ancillary bits k 1 2 ,z 1 2 that the corresponding carry and sum bit. As such, E S (u ∗ ) = 0 ⇔ u ∗ (z 1 1 ) = u ∗ (z 1 2 ) = u ∗ (k 1 2 ) = 0, since the latter condition is equivalent to saying that the bitwise sum of the two sets is zero. This is represented in Fig. 6.3A. Each box in the diagram refers to a generalized full adder. In words, we add the assignments u ∗ 1 ,u ∗ 2 of the bits s 1 1 ,s 1 2 and add the respective carry bit assignment with the assignment u ∗ 3 on the bit s 2 3 . The resulting integer is given by E {z 1 1 ,z 1 2 ,k 1 2 } (u ∗ ({z 1 1 ,z 1 2 ,k 1 2 })) = u ∗ (z 1 1 )+2× u ∗ (z 1 2 )+4× u ∗ (k 1 2 ) - the first row sum bit, the second row sum bit, and what can be considered the third row sum bit added with their respective power of two. This must be zero if u ∗ defines subsets of equal sums and therefore u ∗ must be zero on each of them. The resulting matrix ˜ K M - here we use a tilde to signify that we are in the reduced construction case - that this process defines can be represented as: 116 A B Figure 6.3: Subfigure A shows the reduced embedding of the EQUAL SUBSET SUM instance with the (improper) integer set{1,1,2}. Each box represents a generalized full adder. Each adder describes a corresponding submatrix in the matrix ˜ K M (check Eq. C.8). Subfigure B shows the full embedding of the same instance. 117 ˜ K M = s 1 1 s 1 2 s 2 3 x 1 1 x 2 1 x 3 1 k 1 1 z 1 1 x 1 2 x 2 2 x 3 2 k 1 2 z 1 2 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 (C.8) One can check that if ⃗ µ ∗ = (1,1,− 1,− 1,− 1,0,1,0,0,0,0,0,0), then ˜ K M ⃗ µ ∗ = ⃗ 0. The vector ⃗ µ ∗ defines the assignment u ∗ (S) = {1,1,− 1} - since s 1 ,s 2 ,s 3 are the first three entries of the vectorized form. This defines the two sets A = {s 1 ,s 2 } and B = {s 3 } as a solution to the EQUAL SUBSET SUM problem posed. One can check that ⃗ µ ∗ = (1,− 1,0,0,0,0,0,0,0,0,0,0,0) is also a solution, corresponding to the sets A = {1} and B ={1}. In this reduced construction, we only used the generalized full adder for the significant bitsofeachs j ∈S foragivenbitentryi. Thishelpstogreatlyreducethesizeoftheresulting embedding, but hopefully still conveys the principal idea behind our reduction. While we 118 could write a general protocol on the same principle, it requires a more involved strategy than the one we take. C.3 The Simple Unreduced Case Tosimplifytheconstructionoftheembeddingatthecostofincreasingtheircorrespond- ing size, we follow the same logic as before, but do not prune the insignificant bits. In the unreduced construction, we “compute” the sum bit by bit. Like in the reduced case, the resulting sum bit at the end of each layer corresponds to a bit entry that the sum defined by u ∗ takes - remember, in the end, the value of P a∈A a− P b∈B b was described bitwise by the value u ∗ took on the last sum bit in each layer plus the last layer’s carry bit (e.g. E S (u ∗ (S))=u ∗ (z 1 )+2× u ∗ (z 2 )+4× u ∗ (k 2 )). This remains the same in the unreduced rep- resentation. Note that nonzero sums can have multiple bit representations when the entries can be negative or positive, i.e. 1 = − 1× 1+1× 2 = 1× 1+0× 2, while the sum 0 has only one. In words, we add the bits of each integer in the corresponding bit entry as well as all carry bits from the previous layer to find the total sum of all the integers if none of them had significant bits beyond this layer. Let u ∗ (z i end ) be the last sum bit for any row i. Then the total sum up to the current layer i can be written as 2 i (2× q+s)+ P i j=1 2 j− 1 u ∗ (z j end ) for some q. Then we identify s as the last sum bit of the current layer, u ∗ (z i end ) and q as the net number of carries passed from layer i to i+1. Consider again the simple case we described earlier. We have a (improper) set S = {1,1,2}, such that we can identify each of these three values as s 1 ,s 2 ,s 3 . Refer to Fig. 6.3B to see the resulting diagram that this construction will give. Then s 1 1 ,s 1 2 ,s 1 3 are the first bits of each of these three. We use generalized full adders to add - bit by bit - the values s 1 1 ,s 1 2 ,s 1 3 and feed the resulting carry bits k 1 1 ,k 1 2 to the next layer while z 1 2 takes the value of the lowest bit entry for the total sum of the assignment. In the second layer we add - bit by bit - the values s 2 1 ,s 2 2 ,s 3 3 ,k 1 1 ,k 1 2 and feed the resulting carry bits k 2 1 ,k 2 2 ,k 2 3 ,k 2 4 to the next 119 layer while z 4 2 takes the value of the second lowest bit entry for the total sum. Since the maximum bit entry was given in row two, row three adds only the carry bits k 2 1 ,k 2 2 ,k 2 3 ,k 2 4 , which generates the carry bits k 3 1 ,k 3 2 ,k 3 3 that are subsequently fed into layer four while z 3 4 is the third lowest bit entry for the total sum. Layer four adds the carry values and generates the corresponding carry bits k 4 1 ,k 4 2 as well as the sum bit z 4 2 . Lastly, layer five adds these values and generates the corresponding carry bit k 5 1 as well as the last sum bit z 4 1 . Tocompleteourdescription,eachlayerhasinternalsumvariablesfromeachgeneralized adder. Every line in the diagram corresponds to a variable; variables that are between two boxes are intermediaries, such as all the carry bits except k 5 1 and all the sum bits except the last ones in each layer. For example, layer one has z 1 1 - an intermediate sum bit that is passed from the first generalized adder to the second. This is in contrast to z 1 2 , which is the sum bit of the second generalized adder and is the lowest bit entry for the total sum of the assignment. All carry bits except for the very last one - the one from the single generalized adder in the last row - are intermediaries. Variables that are not between boxes are determined; s j i are set to the j-th lowest bit in the i-th integer of the set S while z i end for layer i and k 5 1 are set to one in the corresponding matrix. C.4 The General Unreduced Case Before we turn our attention to a full protocol for the general unreduced case, we give a more intuitive and visual description of the reduction. Fig. 6.4 gives a schematic of what the general case looks like. Note that Fig. 6.3B fits precisely this description as well. We call the generalized function with the truth table corresponding to Table 6.1 as GA s and GA k for the sum and carry bit respectively, and so the constraints we consid- ered earlier enforce: u ∗ (c) = GA c (u ∗ (a),u ∗ (b)) and u ∗ (s) = GA s (u ∗ (a),u ∗ (b)). We use the common convention of writing u ∗ (a 1 ,...,a k ) as a condensed form of (u ∗ (a 1 ),...,u ∗ (a k )) so that GA c (u ∗ (a),u ∗ (b)) ≡ GA c (u ∗ (a,b)). These are not proper functions since GA s 120 Figure 6.4: This figure shows the layout of the generalized complete adder for enforcing that u ∗ is only valid if the corresponding u on S = {s 1 ,...,s n } is also valid. In each row, a box corresponds to a generalized adder (with the truth table given in Table 6.1) where the output of that whole row (labeled by the sum bit z) is zero if and only if u ∗ is valid. After m (the largest bit length of any s i ∈ S) rows, the next rows are fed only carries from the previous rows. As such, the number of generalized adders decreases by one, until the very last row, where we have that z mn 1 and k mn 1 should both be zero for u ∗ to be valid on the set. The final constraint matrix is a representation of this diagram, with each generalized adder representing a submatrix that enforces the relationship shown in Table 6.1 (check Eq. C.1). and GA k sometimes have two valid modes of operation. We also define GA s (u ∗ (s 1:k )) = GA s (GA s (...GA s (u ∗ (s 1 ),u ∗ (s 2 )),...),u ∗ (s n )) to help condense our writing. To enforce the right bit addition, we use the following protocol: 1. Let l =1,K=Ø, and A=Ø. 121 2. Generate (n− 1)l carry bits (k l 1 ,...,k l (n− 1)l ) and append them to A as well as (n− 1)l sum bits (z l 1 ,...,z l (n− 1)l ) and append them to A. Add n− 1 constraints to K that will enforce GA between s l 1 ,...s l n in order, such that for any assignment u ∗ , u ∗ is valid if and only if u ∗ (k l 1 )=GA k (u ∗ (s l 1 ,s l 2 )) (C.9) u ∗ (z l 1 )=GA s (u ∗ (s l 1 ,s l 2 )) (C.10) u ∗ (k l i )=GA k (u ∗ (z l i− 1 ,s l i+1 ))=GA k (GA s (u ∗ (s l 1:i ))) ∀i∈[2,n− 1] (C.11) u ∗ (z l i )=GA s (u ∗ (z l i− 1 ,s l i+1 ))=GA s (u ∗ (s l 1:i+1 )) ∀i∈[2,n− 1] (C.12) Then place nl constraints to K that will enforce GA between{k l− 1 1 ,...k l− 1 (n− 1)(l− 1) } (the carry ins from the previous layer) and z n− 1 such that for any assignment u ∗ , u ∗ is valid if and only if: u ∗ (k l i )=GA k (u ∗ (z l i− 1 ,k l− 1 i− n ))=GA k (GA s (u ∗ (s l 1:n ,k l− 1 1:i− n ))) ∀i∈[n,(n− 1)l] (C.13) u ∗ (z l i )=GA s (u ∗ (z l i− 1 ,k l− 1 i− n ))=GA s (u ∗ (s l 1:n ,k l− 1 1:i− n )) ∀i∈[n,(n− 1)l] (C.14) 3. Let l =l+1. If l≤ m, then Go to Step 2. 4. At the last run of step 2., we had (n− 1)m total carry bits. Now we add layers feeding carriesforwardlikebefore,butwithoutintroducinganynewbitsfromtheactualintegers. As such, in each layer, we will have one less carry bit generated than the layer before it. 5. Let r =1 6. Generate (n− 1)m− r carry bits{k r+m 1 ,...,k r+m (n− 1)m− r } and append them toA as well as (n− 1)m− r sum bits (z r+m 1 ,...,z r+m (n− 1)m− r ) and append them toA. Add (n− 1)m− r 122 constraints to K that will enforce GA on the carry bits of the previous layer. For the first layer: u ∗ (k m+1 1 )=GA k (u ∗ (k m 1 ,k m 2 ))) (C.15) u ∗ (z m+1 1 )=GA s (u ∗ (k m 1 ,k m 2 ))) (C.16) u ∗ (k m+1 i )=GA k (u ∗ (z m+1 i ,k m i+1 )) ∀i∈[2,m(n− 1)− 1] (C.17) u ∗ (z m+1 i )=GA k (u ∗ (z m+1 i ,k m i+1 )) ∀i∈[2,m(n− 1)− 1] (C.18) For all the subsequent layers: u ∗ (k m+r 1 )=GA k (u ∗ (k m+r− 1 1 ,k m+r− 1 2 ))) (C.19) u ∗ (z m+r 1 )=GA s (u ∗ (k m+r− 1 1 ,k m+r− 1 2 ))) (C.20) u ∗ (k m+r i )=GA k (u ∗ (z m+r i ,k m+r− 1 i+1 ))=GA k (GA s (u ∗ (k m+r− 1 1:i+1 ))) ∀i∈[2,m(n− 1)− r] (C.21) u ∗ (z m+r i )=GA s (u ∗ (z m+r i ,k m+r− 1 i+1 ))=GA s (u ∗ (k m+r− 1 1:i+1 )) ∀i∈[2,m(n− 1)− r] (C.22) 7. Let r =r+1. If r≤ m(n− 1), go to Step 6. 8. Lastly add constraints to force the last sum bit in each row to be zero, those constraints simply are{z l (n− 1)l } for l∈{1,...,m} and{z m+r m(n− 1)− r } for r∈{1,...,m(n− 1)} (check Fig. 6.4). We also add{k m(n− 1) 1 }. Theorem C.1. Suppose there exists u such that P n i=1 u i s i = 0, then and only then does there exist u ∗ such that u ∗ (z l end )=u ∗ (k mn 1 )=0 (where z l end refers to the last sum bit in each row as shown in 6.4). Then E S (u(S))=0⇔E S∪A (u ∗ (S∪A))=0⇔K M ⃗ µ ∗ =0. Proof. We first consider the forward direction. First recognize that P n i=1 u i s i = P n i=1 P m j=1 u i s j i 2 j . It must be that P n i=1 u i s 1 i mod 2 = 0. Then P n i=1 u i s 1 i = σ 1 ∈ {...,− 4,− 2,0,2,4,...}. 123 Note that if inputs a and b have different signs for the generalized adder, the carry and sum bits are both zero, if a and b are the same sign then they pass a carry. When one is zero, then the other one is simply passed on using the primary operation of GA k and GA s . In the forwarddirectionoftheproof,weonlyneedtoconsidertheprimaryoperations. Assuch,itis clearthatu ∗ (z 1 n− 1 )=0sincethenumberofpositiveandnegativeinputsaddediszeromodulo 2. It should also be straight forward to see that P n− 1 i=1 u ∗ (k 1 i ) = σ 1 2 . Now recognize that P n i=1 P l j=1 u i s j i /2 l =σ l ∈{...,− 4,− 2,0,2,4,...}. Notethatσ l = σ l− 1 2 + P n i=1 u i s l i where wecanidentify σ l− 1 2 = P f[l− 1] i=1 u ∗ (k l− 1 i ),withf[x]=Θ( m− x)x(n− 1)+Θ( x− m)(m(n− 1)− x) foralli. Heref[x]hasaHeavisidestepfunctiontodifferentiatebetweentheindexingofrows generated by Step 2 of the protocol versus those generated later by Step 6. Again it is clear thatu ∗ (z l f(l) )=0sincethenumberofpositiveinputsandnegativeinputsof u ∗ (s l 1:n ,k l− 1 1:f(l− 1) ) is zero modulo 2. Since P n i=1 |s i | < n2 m , we must only worry at most about mlog(n) rows, but we have mn rows as zero for u ∗ . We now consider the backward direction. The proof will look very similar to the forward direction, but now we also have to give some consideration that u ∗ could make use ofsecondary operations,notjustprimary operations. Considerinaspecificrow,weusedthe secondaryoperations,e.g. ˜ GA k (1,0)=1and ˜ GA s (1,0)=− 1. Hereweusedthetildetoalert the reader that these are the secondary operations specifically. We know that u ∗ (z l end ) = 0 for any layer l as the assumption, and so the number of ˜ GA operations is even. It must be of opposite kinds such that the number of total carries is unchanged (since they are still valid operations such that 2c+s = a+b) - for every operation that propagrates an extra carry at the expense of reducing its sum bit there must be a secondary operation that reduces its carry bit to surplus its sum bit. If not, then u ∗ (z l end ) ̸= 0. Then we can replace them with the primary operations. The rest of the arguments follow through as before. We have u ∗ (z l n− 1 )=0andsince P n i=1 u ∗ (s 1 i )= P n− 1 i=1 2× u ∗ (k 1 i )+u ∗ (z 1 n− 1 )withu ∗ (z 1 n− 1 )=0, wehave σ 1 = P n− 1 i=1 u ∗ (k 1 i ). Again σ l +u ∗ (z l end )= σ l− 1 2 + P n i=1 u ∗ (s l i ) and we know that u ∗ (z l end )=0. 124 By the same bound, we know that after mn rows having zero on all the outputs is sufficient to see that P n i=1 P m j=1 u ∗ (s j i )=0 and so let u(s i )=u ∗ (s i ) for every s i ∈S. Given an input integer set S, this protocol outputs constraint set K ={K 1 ,...,K S∪A } (the variables with indices such that the entry is nonzero in K M ) such that an assignment vector ⃗ µ ∗ has value zero for every constraint in the set if and only if these exists a valid assignment for S that defines two disjoint nonempty sets A and B such that P a∈A a− P b∈B b = 0. We can then identify the constraint operators as the row vectors of K M as coefficients on spin-z operators on each qubit: ˆ K i = P |S∪A| j k m ij σ z j , such that our solver for 0-1-LP-QCOMMUTE finds a Hermitian matrix that commutes with these constraint operators. C.5 Proof of Runtime In the worse case, we construct no more than 5 new variables and 6 constraints for each GA illustrated in Fig. 6.4. For row i < m+1, this leads to no more than 5(n− 1)i new variables and 6n(n− 1)i constraints. Then after row m, we have no more than 5(n− 1)m 2 variables and 6(n− 1)m 2 constraints. Row i > m has no more than mn− i generalized adders, creating no more than 5(mn− i) new variables and 6(mn− i) constraints. In total, we have no more than O(m 2 n 2 ) variables and O(m 2 n 2 ) constraints. The constraint matrix therefore has sizeO(m 2 n 2 )× O(m 2 n 2 ). The reduction is therefore a polynomial time algorithm. C.6 Reducing a Solution of ILP-COMMUTE to a Solution of EQUAL SUBSET SUM WeconsiderthesamesetupasinSection2.5. UsingtheprotocolfromSection??(check Fig. 6.2) we can reduce any instance of the EQUAL SUBSET SUM problem with polynomial overhead, and if any solution to 0-1-LP-QCOMMUTE exists, there must be v and w (and therefore the sets A and B) to describe at least one off-diagonal term in the spin-z basis. By 125 our construction, the ancilla bits used for forcing are the bits beyond the n-th bit. Similarly, if a solution to EQUAL SUBSET SUM exists, by selecting the values of v and w to match the indicesofchosenelementsforthesetsAandB,weareabletosetthevaluesofthefirst nbits and then propagate their value through the general adder to find v ′ and w ′ over the enlarged space. Then H = N |S S A| i=1 (σ + ) v ′ i (σ − ) w ′ i + N |S S A| i=1 (σ − ) v ′ i (σ + ) w ′ i solves 0-1-LP-QCOMMUTE. This leads to the following Theorem: Theorem C.2. 0-1-LP-QCOMMUTE is NP-Hard. Through the same proof that ILP-QCOMMUTE is polynomial verifiable, 0-1-LP-QCOMMUTE is likewise polynomial verifiable. Theorem C.3. 0-1-LP-QCOMMUTE is NP-Complete. It also leads to an important corollary: Corollary C.3.1. {− 1,0,1}-LP-QCOMMUTE is NP-Complete. As well as another proof to the result in Section 2.5: Corollary C.3.2. ILP-QCOMMUTE is NP-Complete. 126 D ProofoftheMatrixImplementationoftheGeneralized Adder Given inputs a and b, we define the matrix on a,b,s,c,x 1 ,x 2 ,x 3 - with x 1 ,x 2 ,x 3 being intermediating ancillas - as: GA M = a b x 1 x 2 x 3 c s a b 1 1 1 0 0 0 0 1 0 1 1 1 0 0 0 1 1 1 1 0 0 1 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 (D.1) As constraints, we can write it as: GA 1 (a,b,x 1 ,x 2 ,x 3 )=0, (D.2) GA 2 (x 1 ,x 3 ,s,c)=0, (D.3) GA 3 (x 2 ,x 3 ,s,c)=0, (D.4) GA 4 (x 1 ,c)=0, (D.5) GA 5 (x 2 ,c)=0, (D.6) GA 6 (x 3 ,s)=0. (D.7) For every generalized adder in Fig. 6.4 (as described in the protocol we gave in Section C.1), we have a submatrix over the corresponding variables. We give a simple case by case proof that GA M enforces u ∗ to be valid if and only if its entries satisfy 2u ∗ (c)+u ∗ (s)=u ∗ (a)+u ∗ (b) as seen in Fig. 6.1. 127 A constraint is satisfied if and only if the assignment u ∗ over the variables of that constraint sums to zero. Then an assignment satisfies all of them if u ∗ (GA i ) = 0 for all i∈[1,6]. Although checkable through brute force calculations, we give simple arguments for emulation of bit addition step by step: If u ∗ (a) = u ∗ (b) = 1, then and only then do we have u ∗ (c) = 1 and u ∗ (s) = 0. If u ∗ (a) = 1 and u ∗ (b) = 1, then two of the auxillary bits must have an assignment of -1 and one must not. If u ∗ (x 1 ) =− 1 then u ∗ (c) = 1 by GA 4 , but then u ∗ (x 2 ) =− 1 by GA 5 and vice versa. Then u ∗ (x 3 ) = 0, otherwise GA 1 cannot be satisfied. Then u ∗ (s) = 0 as wanted. Suppose that u ∗ (s) = 0 and u ∗ (c) = 1, then likewise GA 4 and GA 3 force that u ∗ (x 1 ) = u ∗ (x 2 ) = − 1. Then from GA 2 , we have that u ∗ (x 3 ) = 0 and so from GA 1 that u ∗ (a)=u ∗ (b)=1. If u ∗ (a) = 1 and u ∗ (b) = 0 or u ∗ (a) = 0 and u ∗ (b) = 1, then and only then do we have u ∗ (c) = 0 and u ∗ (s) = 1 or u ∗ (c) = 1 and u ∗ (s) = − 1. Suppose that u ∗ (a) = 1 or u ∗ (b)=1, but not both. From GA 1 , we know that either one of the auxillary bits must take value -1 or two take the value -1 and one takes the value 1. If either x 1 or x 2 take value -1, but not the other then GA 4 and GA 5 lead to a contradiction, then if x 3 = 1 and by GA 6 it must be that s =− 1. Otherwise x 1 = x 2 = 0 and so u ∗ (x 3 ) =− 1 by GA 2 and GA 3 . GA 6 forces that u ∗ (s) = 1. Suppose that u ∗ (c) = 0 and u ∗ (s) = 1. Then u ∗ (x 3 ) =− 1 from GA 6 and u ∗ (x 1 ) = u ∗ (x 2 ) = 0 from GA 4 and GA 5 . Then GA 1 is only satisfied if u ∗ (a) = 1 or u ∗ (b) = 1, but not both. Suppose instead that u ∗ (c) = 1 and u ∗ (s) =− 1. Then by GA 6 it must be that x 3 = 1. By GA 5 and GA 4 , it must be that x 1 =− 1 and x 2 =− 1. Then by GA 1 it must be that u ∗ (a) or u ∗ (b) is 1, but not both. If u ∗ (a)+u ∗ (b)=0, then and only then do we have u ∗ (c)=0 and u ∗ (s)=0. Suppose thatu ∗ (a)+u ∗ (b)=0. FromGA 1 , weknowthatatmosttwoauxillarybitsarenon-zeroand they have opposite sign. From GA 4 and GA 5 if one of the first two auxillary bits is non-zero thensoistheotherone,buttheymusthavethesamesign. AssuchGA 1 canonlybesatisfied with u ∗ (x 1 ) = u ∗ (x 2 ) = u ∗ (x 3 ) = 0. Then it follows that u ∗ (c) = u ∗ (s) = 0. Suppose that 128 u ∗ (c) = u ∗ (s) = 0. From GA 4 , GA 5 , and GA 6 , we have that u ∗ (x 1 ) = u ∗ (x 2 ) = u ∗ (x 3 ) = 0. Then GA 1 can only be satisfied if u ∗ (a)+u ∗ (b)=0. The same logic works if we swap the values 1 and− 1 everywhere in the above proof. 129
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Leipold, Hannes
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Imposing classical symmetries on quantum operators with applications to optimization
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2023-05
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commutative algebra
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