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Destructive decomposition of quantum measurements and continuous error detection and suppression using two-body local interactions
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Destructive decomposition of quantum measurements and continuous error detection and suppression using two-body local interactions
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DESTRUCTIVE DECOMPOSITION OF QUANTUM MEASUREMENTS AND CONTINUOUS ERROR DETECTION AND SUPPRESSION USING TWO-BODY LOCAL INTERACTIONS by Yi-Hsiang Chen A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (PHYSICS) December 2020 Copyright 2020 Yi-Hsiang Chen Acknowledgements First of all, I would like to thank my advisor Todd Brun. Working with him has been a very special experience in many aspects. Understanding a complex concept through a simple example is one of his talent that inspires me a lot. He could also discuss almost any questions or topics we throw at him, and provide insightful feedback. He has plenty of other traits I really admire. He has been very free in terms of research style but very supportive when I need help in real-life situations. In many ways, I learn a lot from him, and I am indebted to him. I would also like to thank my friend Joshua, who was there with me to share my joys and sorrows when I really needed. I am also appreciated the cheerful time I spent with (another) Joshua and Yating who often hung out with me to nd delicious food in LA. I would thank my other friends who spent time with me and be part of my PhD memory. I want to thank my parents and my brother. Although we are far away apart, they have always had my back. I am truly thankful that we are a family. Last but not least, I am grateful to my wife Hnin, who has made my life more colorful and brought me plenty of happiness. She has been an indispensable part of my PhD life. ii Contents Acknowledgements ii List of Tables v List of Figures vi Abstract viii 1 Introduction 1 2 The lossy measurement model 8 3 POVMs with commuting POVM elements 10 3.1 The construction and its properties . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 The case offj0i;j1ig as a boundary condition . . . . . . . . . . . . . . . . . 15 3.3 Outcome probabilities in the general case . . . . . . . . . . . . . . . . . . . . 16 3.4 The cases with commuting POVM elements . . . . . . . . . . . . . . . . . . 18 4 Generalization to an arbitrary POVM 20 4.1 POVM random walk in a simplex . . . . . . . . . . . . . . . . . . . . . . . . 24 4.2 Constraints by destructive weak measurements . . . . . . . . . . . . . . . . . 29 4.3 The probability to obtain an outcome . . . . . . . . . . . . . . . . . . . . . . 33 5 Continuous indirect measurements 37 5.1 Indirect detections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 5.1.1 Detection methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5.1.2 ZZ example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.2 An application to the four-qubit Bacon-Shor code . . . . . . . . . . . . . . . 47 5.2.1 Error detection and suppression . . . . . . . . . . . . . . . . . . . . . 51 5.3 Constructing the Hamiltonian for indirect detection . . . . . . . . . . . . . . 57 5.3.1 First example: ZZ detection . . . . . . . . . . . . . . . . . . . . . . . 59 5.3.2 Construction for the four-qubit Bacon-Shor code . . . . . . . . . . . . 60 6 Conclusion 64 6.1 Conclusion for the lossy measurement model . . . . . . . . . . . . . . . . . . 64 6.2 Conclusion for continuous indirect measurements . . . . . . . . . . . . . . . 65 iii Reference List 67 Appendix 70 Correspondence between4 ~ r and4 n1 . . . . . . . . . . . . . . . . . . . . . . . . 70 Bayes rule relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 iv List of Tables 5.1 Code basis: bar/un-bar represents encoded/physical basis. . . . . . . . . . . 48 v List of Figures 4.1 The vector~ x, starting from the center, characterizes the position of the walk in4 2 . A 3-outcome measurement is performed at every ~ x. Each outcome corresponds to a small displacement that takes ~ x to a new position ~ x +~ x k . At long times, the walk approaches one of the vertices. . . . . . . . . . . . . 25 4.2 The map from4 2 to a triangle4 ~ r in the Bloch sphere. . . . . . . . . . . . . 29 5.1 (a), (b), (e) and (f) are the estimator approach. The signals from measuring the physical state drive the estimator to the Z 1 Z 2 =1 eigenspace the phys- ical state is in. (a) and (e) are the evolutions ofhZ 1 Z 2 i ^ when is in the +1 eigenspace of Z 1 Z 2 . (b) and (f) are the cases when is in the1 eigenspace ofZ 1 Z 2 . (c), (d), (g) and (h) represent the average function I(t). It converges to 1 when the physical state is in the +1 eigenspace, and it converges to 0 when the state is in the1 eigenspace. . . . . . . . . . . . . . . . . . . . . . 46 5.2 I x;z (t) for the four eigenspaces. The red is I z (t) and the blue is I x (t). The average function converges to 1 (or 0) if the corresponding stabilizer is in the +1 (or1) eigenspace. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.3 The estimator approach. The blue is the evolution ofhS x i ^ . The red is the evolution ofhS z i ^ . It is shown that the estimator approaches the eigenspace the system belongs to. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.4 An X 1 error happened at t = 20 as indicated by the black line. After the error, I z starts to approach 0 andhS z i ^ ips to1. . . . . . . . . . . . . . . 51 5.5 (a) showshS z;x i for the system. After a period, S x is ipped to1 while S z remains at +1. (b) shows I z;x (t). After a period, I x (t) starts to decay while I z (t) remains at +1. (c) showshS z;x i ^ for the estimator.hS x i ^ approaches +1 initially, but it ips to1 afterhS x i!1. . . . . . . . . . . . . . . . . . . 52 5.6 A sample of I x;z (t) with an X mz error at t = 20 and an X 1 error at t = 100. The red curve represents I z (t), and the blue curve represents I x (t). I x (t) remains at 1 because the errors commute withS x . I z (t) ips to1 due to the X mz error and then converges to 0 after the X 1 error happened. The window width w is set to 40=k in this example. . . . . . . . . . . . . . . . . . . . . . 53 vi 5.7 The evolutions of the stabilizers under various error models. (a) is the case for 1/f Hamiltonian noise. (b) is the case for constant Hamiltonian errors. (c) is the case for white noise. They are ensemble averages over 1000 trajectories. 54 5.8 D(t) = 1 2 jj(t)(0)jj 1 . The red curve is the case without measurements while the blue curve is the case with continuous indirect detection. They are ensemble averages over 1000 trajectories. . . . . . . . . . . . . . . . . . . . . 56 5.9 The evolution ofhS z i under an on-resonance X 1 error. The blue curve is the case when the full indirect detection scheme is applied. The red curve is the case when the Hamiltonian for indirect detection is applied but no measurements are made on the monitor qubits. . . . . . . . . . . . . . . . . 57 5.10 The evolution ofhS z i under the full 8-qubit construction. The orange curve includes the indirect detection alone without any errors. The blue curve also includes 1/f Hamiltonian noise. The green curve is the 1/f Hamiltonian noise alone without any indirect measurement. They are ensemble averages over 500 trajectories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 vii Abstract The rst part of this thesis focuses on a study of weak measurements. We consider a qubit model of destructive weak measurements, which is a toy version of an optical cavity, in which the state of an electromagnetic eld mode inside the cavity leaks out and is measured destructively while the vacuum statej0i leaks in to the cavity. At long times, the state of the qubit inevitably evolves to bej0i, and the only available control is the choice of measurement on the external ancilla system. We show that despite the lossy nature of these weak measurements, any POVM can still be achieved by a sequence of these destructive weak measurements. The second part is an application of continuous measurements for stabilizer measure- ments in a quantum code. We provide a method to passively detect the value of a high- weight operator using only two-local interactions and single-qubit continuous measurements. This approach involves joint interactions between the system and continuously-monitored ancillary qubits. The measurement outcomes from the monitor qubits reveal information about the value of the operator. This information can be retrieved by using a numerical estimator or by evaluating the time average of the signals. The interaction Hamiltonian can be eectively built using only two-local operators, based on techniques from perturbation theory. We apply this indirect detection scheme to the four-qubit Bacon-Shor code, where the two stabilizers are indirectly monitored using four ancillary qubits. Due to the fact that the four-qubit Bacon-Shor code is an error-detecting code and that the Quantum Zeno viii Eect can suppress errors, we also study the error suppression under the indirect measure- ment process. In this example, we show that various types of non-Markovian errors can be suppressed. ix Chapter 1 Introduction The original understanding of a quantum measurement is a projective measurement, i.e., it takes a system to an eigenstate of an observable with a certain probability. Let O be an observable with a spectral decomposition O = P i i P i , where i and P i are the eigenval- ues and the eigenprojectors. The set of projectorsfP i g satises P i 0, P i P j = P i ij and P i P i = I, where ij is the Kronecker delta and I is the identity operator. The eect of performing a projective measurement of O is to apply a projector P i to a systemj i with probabilityh jP i j i such thatj i becomes P i j i= p h jP i j i after the measurement. This causes the system to become an eigenstate ofO. This is the so-called wave function collapse of a quantum measurement. However, more recent studies in quantum trajectories and mea- surements elevate the framework to include processes that do not necessarily cause a collapse of a wave function. These measurements consist of a set of operatorsfM i g with the only con- dition P i M y i M i = I. The system, after the measurement, becomes M i j i= q h jM y i M i j i with probabilityh jM y i M i j i. This is called a generalized measurement, which can be real- ized as a system interacts unitarily with an external environment which is then measured. Some information about the system can be obtained via its interaction with the measured environment while the system does not collapse onto an eigenstate. This kind of indirect measurement can exhibit special behavior absent from strong projective measurements. Weak measurements are an interesting class of generalized measurements. A weak mea- surement disturbs the state of the measured system by only a small amount, but also yields only a very small amount of information about the system, on average [1]. A sequence of weak measurements can be used to describe a continuous measurement process. The state of the system evolves stochastically, according to a quantum trajectory equation [7, 43, 24, 18, 14, 13]. There are two types of weak measurements|diusive and jump-like 1 weak measurements. A diusive weak measurement changes a state by a small amount while each measurement outcome happens almost equally likely. A jump-like weak measurement can cause a state to change signicantly, but that can only happen with a small probability. While each weak measurement provides only a small amount of information, a sequence of measurements can allow information to accumulate, so that the system undergoes a strong measurement. Earlier work has shown that any generalized measurement can be decomposed into a series of weak measurements [33]. For a two-outcome measurement, the sequence of weak measurements takes the form of a random walk in one dimension. In [33], it is shown that any two-outcome projective measurement,fP 1 ;P 2 g with P i P j = ij P i and P 1 +P 2 =I, can be decomposed as a weak measurement process using operators P () = r 1 tanh() 2 P 1 + r 1 tanh() 2 P 2 ; where 1. It can be veried that P 2 () +P 2 () = I. At every step, one performs a weak measurement with measurement operatorsfP ();P()g. Suppose the initial state is j i, it can be expressed in terms of the components in the orthogonal spaces, j i =P 1 j i +P 2 j i = p h jP 1 j i P 1 j i p h jP 1 j i + p h jP 2 j i P 2 j i p h jP 2 j i r 1 tanh(x 0 ) 2 j 1 i + r 1 + tanh(x 0 ) 2 j 2 ij (x 0 )i; where x 0 is a real number that characterizes j i's weights in P 1;2 , and j 1;2 i P 1;2 j i= p h jP 1;2 j i. After the rst step, the state becomes j (x 0 )i P ()j (x 0 )i p h (x 0 )jP ()P ()j (x 0 )i = r 1 tanh(x 0 ) 2 j 1 i + r 1 + tanh(x 0 ) 2 j 2 i: 2 Depending on the outcome of either P (), the weak measurement slightly increases/decreases the weight in the space of P 1 . After many steps, the state becomes j (x)i, wherex depends on all the previous outcomes (for example,x =x 0 + + ). The whole measurement process is mapped into a one-dimensional random walk starting at x 0 . When x approaches1 at long times,j (x)i!j 1 i and the outcome of the whole sequence is \1." When x approaches +1,j (x)i!j 2 i the outcome is \2." It can also be shown that the probabilities of obtaining the outcome 1 and 2 are the same ash jP 1 j i and h jP 2 j i correspondingly. Therefore, the projective measurement,fP 1 ;P 2 g, is decomposed into a sequence of weak measurements. This property provides an essential ingredient for proving that any two-outcome gener- alized measurement,fM 1 ;M 2 g, can be decomposed into a sequence of weak measurements. The procedure given in [33] consists of nding the positive operators in the polar decompo- sitions forM 1;2 and performing a weak measurement sequence corresponding to the positive operators. The construction of the weak measurements involves attaching an ancillary qubit to the state, performing a joint unitary operation, and implementing a two-outcome weak measurement on the ancilla followed by another joint unitary that makes the ancilla inde- pendent from the stochastic variable x. Specically, there exists measurement operators ^ M(x;) =U(x) (I P ())U(x) such that ^ M(x;) ((x) j0ih0j) ^ M y (x;)! M(x;)(x)M y (x;) Tr[M(x;)(x)M y (x;)] j0ih0j; for some weak measurement operators M(x;) that only act on . The existence of M(x;), which is explicitly given in [33], guarantees a weak measurement decomposi- tion of the two-outcome generalized measurementfM 1 ;M 2 g. Combining this with the fact that a generalized measurement with an arbitrary outcome can be achieved by performing 3 a sequence of two-outcome measurements, one can deduce that any generalized measure- ment can be decomposed into a sequence of weak measurements. It is also shown in [41] that a generalized measurement can be realized as a continuous stochastic process which corresponds to a random walk in a simplex. For a generalized measurement, the random walk requires feedback|the weak measure- ment done at any stage in the process depends on the outcomes of measurements at earlier stages. However, these constructions assume that one can perform any weak measurement, which requires that one can do any weak joint unitary transformation on the system and ancilla. In practice, this is not always the case, so it makes sense to also look at what kinds of measurements can be achieved with more restrictive families of weak measurements. Ref. [15] studies what measurements can be produced using only a qubit ancilla with a constant inter- action Hamiltonian between the ancilla and the system. It turns out that only measurements with two distinct singular values can be achieve in this setup. Another work [16] examines the class of decomposable generalized measurements using a tunable interaction Hamilto- nian between a stream of qubit probes and the system. The result shows that the allowed Hamiltonians form a nite-dimensional Jordan algebra and they give rise to a large class of measurements. In this thesis, we look at the case of \destructive" measurements where the system is projected onto a certain state after the measurement, while the probabilities of the outcomes re ect a certain POVM. A simple example of this type of measurements is the two-outcome qubit measurement consisting of M 1 =j0ih+j and M 2 =j0ihj. It projects the system onto thej0i state for both outcomes while performing a POVM in the x-basis. A physical example of a destructive measurement is photodetection. In photodetection, the observer waits for photons to be absorbed by the detector. After a long monitoring time, all photons are consumed and the source system is projected onto the vacuum state. If we continuously monitor this detection process, the outcomes give a stochastic evolution for the process of photodetection. 4 Motivated by this example, we developed a model of destructive weak measurements for qubits as described in Chapter 2. It consists of a weak unitary swap between the system and the ancilla qubit, in the initial statej0i, followed by a projective measurement on the ancilla qubit. The weak swap between the system and thej0i state is the same for each step and the cumulative eect of the swap leads to a projection ontoj0i for the system at long times. This is analogous to photodetection, in which photons must be absorbed to be detected: the source system is projected onto the vacuum state|the analog ofj0i|after all photons are consumed. Note that the destructiveness in this toy model does not come from the projective measurement on the ancilla, but the constant weak swap interaction between the system and the ancilla qubits and the fact that the ancillas begin in the statej0i at each step. Although the model is destructive, it is shown in Chapter 3 that it can achieve any POVM with commuting POVM elements, including all projective measurements on a qubit. However, general POVMs can not be done by this method. In Chapter 4, we develop another method, the random walk in a simplex, to achieve an arbitrary POVM with any number of outcomes. In the second part of this thesis, we introduce an application of the well-studied continu- ous measurement [44, 23], which can also be described as a sequence of weak measurements. A continuous measurement of an observableX is a detector with a continuous output current I(t) = R dI such that dI =hXidt +dW=2 p , where dW , representing the measurement noise, is a Gaussian distributed random variable with zero mean and the variance dt. is the measurement rate. The detector causes a system to change by a measurement operator M(dI) = 2 dt 1 4 e (dIXdt) 2 dt ; given the outcome is dI. When the system is in an eigenstate of the observable X with eigenvalue x, the probability density of obtaining the result dI is h jM y (dI)M(dI)j i = 2 dt 1 2 e 2 (dIxdt) 2 dt ; 5 which is exactly a Gaussian distribution with mean xdt and variance dt. One can check that the above probability density integrates to 1 over dI. The instantaneous measurement outcome dI contains information of the expectation value of X, but the value of dI is dom- inated by the noise dW , which has a magnitude of order p dt. This means the information gain at every time step is small. However, the gain of information accumulates as the noise averages out with time. This type of continuous measurement for single-qubit observables has been well studied both theoretically and experimentally [27, 28, 42, 31, 12, 38, 37]. However, methods to perform practical continuous measurement of multi-qubit observables have not been fully developed. In the context of continuous quantum error correction, early work by Paz and Zurek [35] introduces a jump-like error correction process, where the recovery operation is applied with probability dt at each time stepdt with rate . This continuous-time jump-like error correction process can be realized as applying a sequence of weak measurements [32], and the minimum number of the required ancillary qubits is found to be nk + 1 for an [[n;k;d]] code [20]. Another framework proposed by Ahn, Doherty, and Landahl (ADL) [2] uses continuous measurements with feedback control to maintain the delity of an unknown quantum state. Some feedback-based error correcting protocols related to the ADL scheme are studied in [3, 4, 39, 8]. A major practical diculty of almost all continuous quantum error correction schemes is that they assume that it is possible to continuously measure multi-qubit operators. Mea- suring high-weight operators is crucial for many quantum codes in the stabilizer formalism [19]. The surface code [6, 17], for example, has stabilizer generators of weight four, and other codes can have even higher-weight stabilizers. To continuously measure these high-weight operators is challenging, since it requires Hamiltonians with many-body terms. In Chapter 5, we introduce a method to indirectly detect the value of a high-weight operator using local two-body interactions and single-qubit continuous measurements. The approach involves applying an interaction Hamiltonian for the system and additional qubits 6 that are being continuously monitored. The information of the system's value for the observ- able translates into dierent signatures of the monitored qubits, which we can identify. The setup can be applied to quantum codes where we detect errors by measuring the stabilizers. As an example, we apply this detection scheme to the four-qubit Bacon-Shor code, which is an error-detecting code that can detect errors by measuring only two-local operators [5]. One major advantage of using this subsystem code [29] is that measuring two-local operators is much easier than measuring higher-weight operators. This advantage also translates into a smaller energy cost in the perturbative construction using two-body interaction described in Chapter 5.3. In this paper, we focus on the measurements of the weight-four stabilizers in the error-detecting Bacon-Shor code. It is well-known that the Quantum Zeno Eect can freeze a state in an eigenstate of an observable that is frequently measured. Since the four-qubit Bacon-Shor code is an error- detecting code, we examine whether errors can be suppressed when we apply the continuous indirect measurement of the stabilizers. In [34], it is shown that non-Markovian errors can be suppressed by the Zeno eect when the system is being directly measured. Our results for indirect detection also agree with this observation. 7 Chapter 2 The lossy measurement model In our model of a lossy measurement device, our system is a single qubit in an initial state j i, which interacts successively with a stream of ancilla qubits all prepared in the initial statej0i. The interaction is a weak swap between the system qubit and the ancilla qubit, followed by a projective measurement of the ancilla. The weak swap is given by U =I cosiS sin; (2.1) where I is the identity and S is the swap operator that exchanges the states of the system and the ancilla: Sj i j0i =j0i j i: (2.2) The parameter controls the strength of the measurement, and is assumed to be small. We then do a projective measurement on the ancilla. If the projectors on the ancilla are E k =je k ihe k j, where P k E k =I, the net eect will be a weak measurement on the state: j ih j! Tr anc h I E k U j ih j j0ih0j U y I E k i Tr h I E k U j ih j j0ih0j U y I E k i = M k j ih jM y k h jM y k M k j i ; (2.3) where the operators M k =he k j0i cosIi sinj0ihe k j (2.4) are the eective measurement operators on the state, and P k M y k M k =I. After each step, the ancilla qubit is discarded and replaced by a fresh qubit in the statej0i. 8 In this lossy model, it is evident that the state approaches thej0i state at long times. However, one can ask if the sequence of measurements can be made equivalent to an arbitrary POVM? In Chapter 3, we show that any POVM with commuting POVM elements can be achieved by a sequence of the lossy weak measurements. In Chapter 4, we generalize the result to all POVMs, which requires a much more complicated formalism. 9 Chapter 3 POVMs with commuting POVM elements This chapter is based on the work in the paper [9]. By choosing dierent projective measurements on the ancilla, one can perform a series of weak measurements on the system such that the whole set of measurements corresponds to a projective measurement. For an arbitrary two-outcome projective measurement with projectorsjb 0 ihb 0 j andjb 1 ihb 1 j on the system, with initial statej i, one can nd a series of projectors E k i on the ancilla, where k i = 0; 1, such that the product of the whole set of measurement operatorsM k 1 k N M k 1 acts as the desired projective measurement, followed by a unitary rotation that moves the basisfjb 0 i;jb 1 ig tofj0i;j1ig. (The subscript k 1 k i indicates that theith measurement operator depends on the whole previous outcomes.) The procedure is as follows. 3.1 The construction and its properties We will use the polar decomposition for the measurement operators: M k =U k q M y k M k : We choose projectors E k =je k ihe k j on the ancilla such that M k =U k p 0;k jb 0 ihb 0 j + p 1;k jb 1 ihb 1 j ; k = 0; 1; (3.1) 10 where 0;k and 1;k are the eigenvalues for the POVM elements: M y k M k = 0;k jb 0 ihb 0 j + 1;k jb 1 ihb 1 j: (3.2) We can always nd projectorsE k on the ancilla such that Eq. (3.2) is satised for any given measurement basisfjb 0 i;jb 1 ig. If the measurement basis isfjb 0 i = aj0i +be i j1i;jb 1 i = bj0iae i j1ig, where a;b 0, a 2 +b 2 = 1 and is real, then the basisfje 0 i;je 1 ig for the projectors E 0 ;E 1 on the ancilla should be chosen as je 0 i;je 1 i = 8 > > < > > : q 1 2 j0i q 1 2 e i j1i if ab; q 1 2 j0i q 1 2 e i j1i if ab; (3.3) where = s 1 4a 2 b 2 1 4a 2 b 2 cos 2 and = +=2. The eigenvalues for the POVM elements in Eq. (3.2) will be 0;0 = 1;1 = 1 2 1 + p 1 4a 2 b 2 cos 2 + sin 2 p 1 4a 2 b 2 cos 2 ; 0;1 = 1;0 = 1 2 1 + p 1 4a 2 b 2 cos 2 sin 2 p 1 4a 2 b 2 cos 2 : (3.4) The resulting unitary in Eq. (3.1) then becomes U k =M k 1 p 0;k jb 0 ihb 0 j + 1 p 1;k jb 1 ihb 1 j ; (3.5) where M k is the operator in Eq. (2.4), when 0;1;k 6= 0, which is true forfjb 0 i;jb 1 ig 6= fj0i;j1ig. By this construction, the rst weak measurement for a projective measurementjb 0 ihb 0 j andjb 1 ihb 1 j on a statej i =jb 0 i +jb 1 i would be M k 1 =U k 1 ( p 0;k 1 jb 0 ihb 0 j + p 1;k 1 jb 1 ihb 1 j); k 1 = 0; 1: (3.6) 11 Depending on the outcomek 1 from the rst measurement, the basis vectorsjb 0 i andjb 1 i will be moved tojb k 1 0 i = U k 1 jb 0 i andjb k 1 1 i = U k 1 jb 1 i. The state after the rst measurement will take the form j k 1 i =M k 1 j i= p p k 1 = 1 p p k 1 p 0;k 1 jb k 1 0 i + p 1;k 1 jb k 1 1 i ; (3.7) where p k 1 is the probability of outcome k 1 : p k 1 =h jM y k 1 M k 1 j i =jj 2 0;k 1 +jj 2 1;k 1 : (3.8) Because the measurement basis changes fromjb 0;1 i tojb k 1 0;1 i, the second set of measurement operators are M k 1 k 2 =U k 1 k 2 p 0;k 1 k 2 jb k 1 0 ihb k 1 0 j + p 1;k 1 k 2 jb k 1 1 ihb k 1 1 j ; (3.9) k 1;2 = 0; 1, where 0;k 1 k 2 and 1;k 1 k 2 are the two eigenvalues of the POVM elementM y k 1 k 2 M k 1 k 2 given that the rst outcome is k 1 . After the second measurement, the state becomes j k 1 k 2 i = 1 p p k 2 jk 1 M k 1 k 2 j k 1 i = 1 p p k 2 jk 1 p k 1 M k 1 k 2 M k 1 j i = 1 p p k 1 k 2 p 0;k 1 0;k 1 k 2 jb k 1 k 2 0 i + p 1;k 1 1;k 1 k 2 jb k 1 k 2 1 i ; (3.10) wherep k 1 k 2 is the probability of obtaining the outcomesk 1 andk 2 , andjb k 1 k 2 0;1 i =U k 1 k 2 U k 1 jb 0;1 i is the measurement basis after the rst two measurements. By continuing the process N times, the state becomes j k 1 k N i = 1 p p k 1 k N M k 1 k N M k 1 j i = 1 p p k 1 k N p 0;k 1 0;k 1 k N jb k 1 k N 0 ihb 0 j + p 1;k 1 1;k 1 k N jb k 1 k N 1 ihb 1 j j i = 1 p p k 1 k N p 0;k 1 0;k 1 k N jb k 1 k N 0 i + p 1;k 1 1;k 1 k N jb k 1 k N 1 i ; (3.11) 12 where p k 1 k N is the probability of getting the string of outcomes k 1 k N : p k 1 k N =jj 2 0;k 1 0;k 1 k N +jj 2 1;k 1 1;k 1 k N : (3.12) As the number N increases, the measurement basisjb k 1 k N 0;1 i approaches toj0i;j1i. This can be proved by looking at the form of the sequence of measurementsM k 1 k N M k 1 . Each measurement operator M k 1 k i is a 2 2 matrix of the form M k 1 k i = he k i j0ie i ihe k i j1i sin 0 he k i j0i cos ! : (3.13) Without loss of generality, one can choosehe k i j0i to be a positive real number with the projectorje k i ihe k i j on the ancilla. Furthermore, if the initial projective measurementjb 0;1 i is notj0i;j1i thenhe k i j0i will not be zero at any point in the process. This can be shown as follows: assume that one starts from a projective measurementfjb 0 i;jb 1 ig6=fj0i;j1ig (that is, neitherjb 0 i = j0i norjb 0 i = j1i). Now suppose that at the ith measurement one getshe k i j0i = 0. This implies that the measurement operator for the ith measurement is M k i = i sinj0ih1j, which means that the (i 1)th measurement is already in the fj0i;j1ig basis. One can easily show that the (i 2)th measurement then must also be in thefj0i;j1ig basis, and so on. Hence, one can deduce that for the rst measurement fjb 0 i;jb 1 ig =fj0i;j1ig. This contradicts the assumption, and therefore nohe k i j0i can be zero throughout the whole process. This also shows that the measurement basis will never befj0i;j1ig at any point in the process if one starts fromfjb 0 i;jb 1 ig6=fj0i;j1ig, though the measurement basis will approachfj0i;j1ig asymptotically. If one performs such a sequence of measurements, the product of the measurement oper- ators becomes M k 1 k N M k 1 = 0 @ 1 x 0 1 A N Y i=1 he k i j0i; (3.14) 13 where x =ie iN sin N X j=1 he k j j1i he k j j0i e i(Nj) cos j1 (3.15) and =e iN cos N : (3.16) From the second line in Eq. (3.11), the measurement basisjb k 1 k N 0;1 i afterN measurements is the eigenbasis of the matrix M k 1 k N M k 1 M y k 1 M y k 1 k N = 0;k 1 0;k 1 k N jb k 1 k N 0 ihb k 1 k N 0 j + 1;k 1 1;k 1 k N jb k 1 k N 1 ihb k 1 k N 1 j / 0 @ 1 x 0 1 A 0 @ 1 x 0 1 A y = 0 @ 1 +jxj 2 x x jj 2 1 A : (3.17) Letjv + i;jv i be the two eigenvectors of the matrix in Eq. (3.17), and denejv + i to be the eigenvector that is closer toj0i. The squared magnitude of the inner product ofjv + i with j0i is jhv + j0ij 2 = 1 +jxj 2 jj 2 + p (1 +jxj 2 +jj 2 ) 2 4jj 2 2 1 +jxj 2 jj 2 + p (1 +jxj 2 +jj 2 ) 2 4jj 2 2 + 4jxj 2 jj 2 : (3.18) In the large N limit,jj 2 goes to 0 like cos 2N , andjhv + j0ij 2 approaches 1 no matter what valuejxj 2 takes. This can be shown as follows. Ifjxj 2 approaches 0, thenjhv + j0ij 2 1jxj 2 jj 2 . Ifjxj 2 approaches innity, thenjhv + j0ij 2 1 (1=jxj 2 )jj 2 . Ifjxj 2 approaches a nite number, thenjhv + j0ij 2 1 (jxj 2 =(1 +jxj 2 ) 2 )jj 2 . By the orthogonality of the eigenbasis,jhv j0ij 2 therefore approaches 0. Hence, the measurement basisjb k 1 k N 0;1 i, which is equivalent tojv i, must approachfj0i;j1ig. By looking at the quantityjhb k 1 k N 0 j0ij 2 , ifjhb k 1 k N 0 j0ij 2 ! 1 we conclude that the outcome isjb 0 i, while ifjhb k 1 k N 0 j0ij 2 ! 0 we conclude that the outcome isjb 1 i. We will now show 14 that the probability of concluding the outcome to bejb 0 i (orjb 1 i) approachesjj 2 (orjj 2 ) in the large N limit. 3.2 The case offj0i;j1ig as a boundary condition Let us rst consider the case where the projective measurement to be performed on the state isjb 0 i =j0i,jb 1 i =j1i. By choosing the projectorsje 0 i =j0i;je 1 i =j1i, the rst measurement operators on the state will be M 0 =e i j0ih0j + cosj1ih1j; (3.19) and M 1 =i sinj0ih1j: (3.20) If one gets outcome 0, then the new measurement basis remains the same:jb 0 0 i =j0i and jb 0 1 i =j1i. The initial statej i =j0i +j1i becomes j 0 i = 1 p p 0 e i j0i + cosj1i ; (3.21) where p 0 =jj 2 +jj 2 cos 2 is the probability of outcome 0. One then repeats the same process for the second measurement. If at any point one gets the outcome 1, then the new measurement basis is ipped,jb 1 0 i =j1i andjb 1 1 i =j0i, and the system state goes toj0i. The process can be terminated at that point, because any subsequent measurements will getj0i on the ancilla with probability 1. Then we conclude the outcome is 1. If one never gets the resultj1i on the ancilla, then we conclude the outcome is 0. If one repeats the measurements N times then the probability of concluding that the outcome is 0 (namely, never gettingj1i on the ancilla), isjj 2 +jj 2 cos 2N , and the probability of concluding that the outcome is 1 is 1 (jj 2 +jj 2 cos 2N ) =jj 2 (1 cos 2N ). In the large N limit, the probability of concluding that the outcome isj0i orj1i approachesjj 2 orjj 2 , respectively. 15 This accomplishes a projective measurementfj0i;j1ig on the state, and this special case sets a boundary condition for the model. 3.3 Outcome probabilities in the general case We now show that the probabilities of the two outcomes match the usual Born rule for a general projective measurement in the basisfjb 0 i;jb 1 ig. For the initial statej i = jb 0 i + jb 1 i, we dene the functionP (jb 0;1 i;) to be the probability of concluding that the outcome isjb 0 i. This is equivalent to the condition thatjhb k 1 k N 0 j0ij 2 ! 1 in the large N limit. P does not depend on the relative phase between and or the relative phase in basis jb 0;1 i, because only the magnitude of and a enter the expression for the probabilities for outcomes as shown in Eq. (3.4) and Eq. (3.12). From probability theory we have the following equation: P (jb 0;1 i;) = p 0 P jb 0 0;1 i; p 0;0 p p 0 ! +p 1 P jb 1 0;1 i; p 0;1 p p 1 ! ; (3.22) where p 0 is the probability of getting outcome 0 from the rst measurement and P jb 0 0;1 i; p 0;0 =p 0 is the probability to conclude the outcome isjb 0 0 i starting from the new state and measurement basis after the rst measurement. If we iterate this expression for N times, it becomes P (jb 0;1 i;) =p 0 P jb 0 0;1 i; p 0;0 p p 0 ! +p 1 P jb 1 0;1 i; p 0;1 p p 1 ! =p 0 " p 0j0 P jb 00 0;1 i; p 0;0 0;00 p p 00 +p 1j0 P jb 01 0;1 i; p 0;0 0;01 p p 01 # +p 1 " p 0j1 P jb 10 0;1 i; p 0;1 0;10 p p 10 +p 1j1 P jb 11 0;1 i; p 0;1 0;11 p p 11 # . . . = X k 1 k N =0;1 p k 1 k N P jb k 1 k N 0;1 i; p 0;k 1 0;k 1 k N p p k 1 k N : (3.23) 16 From the special case above, we have P (j0; 1i;) =jj 2 +jj 2 cos 2N , and jP (j0; 1i;)jj 2 j cos 2N : (3.24) By the continuity of the function P (jb 0;1 i;) atjb 0;1 i =j0; 1i, there exists a quantity > 0 such that jP (jb 0;1 i;)P (j0; 1i;)j; (3.25) and asjb 0;1 i!j0; 1i, ! 0. Now we can evaluate the function P in Eq. (3.23). From the result given earlier, the measurement basisjb k 1 k N 0;1 i approachesj0; 1i for any outcomes k 1 k N in the large N limit. By combining Eq. (3.24) and Eq. (3.25), the function P in Eq. (3.23) has the following bound: P jb k 1 k N 0;1 i; p 0;k 1 0;k 1 k N p p k 1 k N jj 2 0;k 1 0;k 1 k N p k 1 k N (k 1 k N ) + cos 2N ; (3.26) where (k 1 k N ) is the quantity in Eq. (3.25) for a certain combination of outcomes k 1 k N . With the expression (3.26), Eq. (3.23) becomes P (jb 0;1 i;) = X k 1 k N =0;1 p k 1 k N P jb k 1 k N 0;1 i; p 0;k 1 0;k 1 k N p p k 1 k N X k 1 k N =0;1 jj 2 0;k 1 0;k 1 k N +p k 1 k N h max(k 1 k N ) + cos 2N i ! =jj 2 + max(k 1 k N ) + cos 2N ; (3.27) where max(k 1 k N ) =(k 1 k N ) is the maximum variation, which occurs for a certain string of outcomes k 1 k N . The last equality in (3.27) is due to Eq. (3.12): 1 = X k 1 k N =0;1 p k 1 k N =jj 2 X k 1 k N =0;1 0;k 1 0;k 1 k N +jj 2 X k 1 k N =0;1 1;k 1 1;k 1 k N (3.28) 17 is satised for all and . Hence, X k 1 k N =0;1 0;k 1 0;k 1 k N = X k 1 k N =0;1 1;k 1 1;k 1 k N = 1: (3.29) All (k 1 k N ) and cos 2N approach zero in the large N limit, and therefore P (jb 0;1 i;) approachesjj 2 . This shows that by using the lossy device, one can perform any projective measurement on a state. 3.4 The cases with commuting POVM elements We have shown that the destructive model can implement an arbitrary projective measure- ment. There is a simple way to perform a more general POVM with any set of mutually commuting POVM elements, by introducing a random number generator together with the lossy device and doing an extra classical postprocessing step. Because commuting POVM elements share the same eigenvectors, they can be expanded in the same eigenbasis. We perform a projective measurement in this eigenbasis, followed by a probabilistic output step. For simplicity, consider a POVM with three commuting POVM elements: E 1 =ajb 0 ihb 0 j +bjb 1 ihb 1 j; E 2 =cjb 0 ihb 0 j +djb 1 ihb 1 j; E 3 = (1ac)jb 0 ihb 0 j + (1bd)jb 1 ihb 1 j; (3.30) where 1a;b;c;d 0 and 1a +c; 1b +d. The eigenbasis isfjb 0 i;jb 1 ig. For an initial statej i =jb 0 i +jb 1 i, the probabilities of outcomes 1, 2, and 3 are P 1 =ajj 2 +bjj 2 ; P 2 =cjj 2 +djj 2 ; P 3 = (1ac)jj 2 + (1bd)jj 2 : (3.31) 18 One rst performs ajb 0 i;jb 1 i projective measurement using the lossy model in the previous section. One then generates the output using a random number generator conditioned on the outcome of the projective measurement. For projective measurement outcome jb 0 i, we output 1, 2, or 3 with probabilities a, c, and 1 a c, respectively; for projective measurement outcomejb 1 i, we output 1, 2 or 3 with the probabilities b, d, or 1bd, respectively. Then the unconditioned probabilities of getting outcomes 1, 2, and 3 will be the same as Eq. (3.31). The outcome of the projective measurement and the values of the random numbers are discarded. One can think of the lossy projective measurement process and the random number generator together as parts of a single device: the outcome from the projective measurement is input into the random number generator, which gives the nal outcome 1, 2, or 3. One can easily generalize this approach into a POVM with an arbitrary number of commuting POVM elements. This shows that any POVM with commuting POVM elements can be achieved by a sequence of the lossy measurements. 19 Chapter 4 Generalization to an arbitrary POVM This chapter is based on the work in the paper [10]. In this section, we show that any POVM can be performed using the lossy measure- ment model. The generalization to an arbitrary POVM requires more complicated methods including pre- and post-processing and a mapping to a random walk in a simplex. To begin, we rst present some general properties of a POVM. Denition 1 (Linearly independent POVM). We say that a POVM,fE i g n i=1 , is linearly independent if P n i=1 E i =I, where E i 's are positive-semidenite operators, and P n i=1 c i E i = 0 has no nontrivial solutions for thec i 's. We denote such a POVM as a linearly independent POVM (LIPOVM). A POVM, in general, can have any number of outcomes. However, it can be decomposed into a linear combination of linearly independent POVMs in the following sense. Proposition 1. Performing a POVM is equivalent to randomly choosing, with certain prob- abilities, to perform one LIPOVM from a collection of LIPOVMs. We provide the following decomposing process to illustrate Proposition 1. Consider a POVM,fE i g n i=1 , with linearly dependent POVM elements. The E i 's are positive-semidenite operators with P n i=1 E i =I, and P n i=1 c i E i = 0 has a nontrivial solution for thec i 's. We use a labeling convention such thatc 1 c 2 c n , wherec 1 ; ;c m 0 and c m+1 ; ;c n < 0. Note that c 1 > 0 and c n < 0 are always true, since otherwise they contradict the positivity of the E i 's. We can construct two POVMs, E A j c 1 c j c 1 E j ; j = 2; ;n (4.1) E B j c n c j c n E j ; j = 1; ;n 1: (4.2) 20 One can easily check that P n j=2 E A j =I and P n1 j=1 E B j =I. We then performfE A j g n j=2 with probability P A = c 1 =(c 1 c n ) and performfE B j g n1 j=1 with probability P B =c n =(c 1 c n ). The probability of obtaining each outcome can be checked to agree with the original value TrE i for a given state . For outcome n, P A Tr E A n = c 1 c 1 c n Tr c 1 c n c 1 E n = Tr [E n ]: (4.3) For outcome 1, P B Tr E B 1 = c n c 1 c n Tr c n c 1 c n E 1 = Tr [E 1 ]: (4.4) For the outcomes i, 2in 1, P A Tr E A i +P B Tr E B i = c 1 c 1 c n Tr c 1 c i c 1 E i + c n c 1 c n Tr c n c i c n E i = Tr [E i ]: (4.5) This process reduces an n-outcome POVM into two (n 1)-outcome POVMs with pre- processing. If either offE A j g n j=2 andfE B j g n1 j=1 is still linearly dependent, one can repeat the same process to reduce the number of outcomes until all POVMs are linearly independent. Hence, an n-outcome linearly dependent POVM is equivalent to randomly choosing from a set of LIPOVMs with certain probabilities. Since a qubit POVM has elements that are 2 by 2 matrices, any qubit POVM with more than 4 outcomes can always be decomposed into 4-or-fewer-outcome LIPOVMs by pre-processing. Denition 2 (Projective POVM). We call a POVM with each element proportional to a projector a projective POVM (PPOVM), i.e., allfE i g n i=1 have the form E i =a i jiihij, where the a i 's are positive. Proposition 2. Performing any qubit POVM is equivalent to performing a projective POVM followed by outputting an outcome with a probability that depends on the result of that pro- jective POVM. 21 Proof. Given a qubit POVMfE i g n i=1 , each E i can be written as E i =a i ja i iha i j +b i jb i ihb i j; (4.6) wherea i ;b i andja i i;jb i i are the eigenvalues and eigenvectors of E i . Leta i b i for alli. We can rewrite each E i as E i = (a i b i )ja i iha i j +b i I; (4.7) sinceja i iha i j +jb i ihb i j =I for each i. Because P n i=1 E i =I, we have I = n X i=1 a i b i 1 P n j=1 b j ja i iha i j: (4.8) Dene a PPOVMf ~ P i g n i=1 as ~ P i a i b i 1 P n j=1 b j ja i iha i j: (4.9) We can rewrite E i in terms of the ~ P i 's; i.e., E i = (a i b i )ja i iha i j +b i n X k=1 a k b k 1 P n j=1 b j ja k iha k j = n X k=1 2 4 ik (a k b k ) +b i a k b k 1 P n j=1 b j 3 5 ja k iha k j = n X k=1 " ik 1 n X j=1 b j ! +b i # ~ P k n X k=1 p(ijk) ~ P k ; (4.10) where ik is the Kronecker delta and p(ijk) ik 1 n X j=1 b j ! +b i : (4.11) 22 One can quickly check that P n i=1 p(ijk) = 1 for any outcome k. Hence, the function p(ijk) can be interpreted as a conditional probability, in the sense that we rst performf ~ P i g n i=1 and then output the nal outcome i with probability p(ijk) when we got result ~ P k fromf ~ P i g n i=1 . Therefore, performing a qubit POVMfE i g n i=1 is equivalent to performing a projective POVM f ~ P i g n i=1 followed by post-processing as dened in Eq. (4.11). Remark. If the qubit POVMfE i g n i=1 is linearly independent, so isf ~ P i g n i=1 . This is because the transformation in Eq. (4.10) has an inverse. That is ~ P i = n X j=1 p 1 ij E j ; (4.12) where p 1 ij = ( ij b i )=(1 P n k=1 b k ). If the set of vectorsfE i g n i=1 is linearly independent, thenf ~ P i g n i=1 will be linearly independent as well. Combining the above properties, we have the following result: any qubit POVM is equiv- alent to pre- and post-processing steps with a set of linearly independent and projective POVMs. That is, for a given qubit POVM, one uses a random number generator to choose, with probabilities given in Proposition 1, a linearly independent POVM, and uses Proposi- tion 2 to nd and perform the projective POVM corresponding to that linearly independent POVM; nally, one outputs the nal outcome with probability p(j), dened in Eq. (4.11). The probability of outputting the nal outcome i will be equal to TrE i , which is the probability of getting outcome i from the original POVM. Note that the only actual measurement made in this process is the linearly independent and projective POVM (LIPPOVM). Hence, for a qubit system, if one can perform any LIPPOVM then all POVMs can be accomplished. We claim that the model of lossy weak measurements can achieve any qubit LIPPOVM and hence any qubit POVM. The following sections will verify the assertion. 23 4.1 POVM random walk in a simplex In earlier work [40], it was shown that a generalized measurement can be decomposed into a series of stochastic processes such that the trajectory corresponds to a random walk in a simplex. Here we use a similar approach to parametrize the measurement operator during the sequence of weak measurements. The intuition is to map the current POVM element at each step of a sequence of weak measurements into a convex combination of the original POVM elements. We will show that, for a linearly independent POVM, the conditions to perform a measurement where each outcome corresponds to a vector in the simplex can be satised. To decompose a linearly independent POVM into stochastic processes of weak measure- ments, we use a family of measurement operators M(~ x) of the form M(~ x) =f(~ x)U(~ x) v u u t n X i=1 x i E i ; (4.13) where U(~ x) is a unitary operator, f(~ x) is a normalization factor, and ~ x24 n1 = ( (x 1 ; ;x n )j n X i=1 x i = 1; x i 08i ) : (4.14) The form of Eq. (4.13) is fairly general because any matrix allows for a polar decomposition, and Eq. (4.13) only requires the positive-semidenite matrix to be the square root of a linear combination of the POVM elements. The vector~ x characterizes the position of the random walk in the (n 1)-simplex,4 n1 . Note that for a given position ~ x in the walk, M y (~ x)M(~ x)/ n X i=1 x i E i : (4.15) This form characterizes how the POVM element of the weak measurement evolves as a convex combination of the original POVM elements. It will be shown in Sec. 4.3 that~ x will approach one of the vertices of4 n1 after many measurements: ~ x! ( ; 0; 1; 0; ). The 24 Figure 4.1: The vector ~ x, starting from the center, characterizes the position of the walk in 4 2 . A 3-outcome measurement is performed at every ~ x. Each outcome corresponds to a small displacement that takes~ x to a new position~ x+~ x k . At long times, the walk approaches one of the vertices. vertex corresponds to the outcome E j of this POVMfE i g n i=1 . An example of performing a 3-outcome POVM is shown in Fig. 4.1. Before making any measurement,~ x is at the middle of4 n1 , so the random walk starts from ~ x =~ x 0 = 1 n 0 B B B @ 1 . . . 1 1 C C C A ; (4.16) andM(~ x 0 ) =U(~ x 0 ) =I. After each weak measurement is made, the new position~ x remains in4 n1 if the weak measurement, with measurement operatorsfM k (~ x)g n k=1 satisfying n X k=1 M y k (~ x)M k (~ x) =I; (4.17) satises M y (~ x)M y k (~ x)M k (~ x)M(~ x)/M y (~ x k )M(~ x k ) (4.18) for each outcome k such that ~ x k 24 n1 : (4.19) 25 Note that ~ x k is a new position in the simplex4 n1 after outcome k, and ~ x k ~ x k ~ x is expected to be a small quantity given that we are performing weak measurements. Using Eq. (4.13) for M(~ x), we can rewrite Eq. (4.18) as v u u t n X i=1 x i E i U y (~ x)M y k (~ x)M k (~ x)U(~ x) v u u t n X i=1 x i E i / n X i=1 (~ x k ) i E i : (4.20) Dene a new set of POVM elements by E i (~ x)U(~ x)E i U y (~ x); 8 i2f1; ;ng: (4.21) It can be easily checked thatfE i g n i=1 is linearly independent if and only iffE i (~ x)g n i=1 is linearly independent. Inserting Eq. (4.21) back into Eq. (4.20), we have M y k (~ x)M k (~ x)/ n X i=1 x i E i (~ x) ! 1 2 n X i=1 (~ x k ) i E i (~ x) ! n X i=1 x i E i (~ x) ! 1 2 : (4.22) We can expand all the operators as linear combinations of Pauli operators and the iden- tity; i.e., for each i, E i (~ x) =q i (I +~ v i ~ ); (4.23) where~ is the vector of Pauli matrices andq i ,~ v i are the expansion coecients. By assump- tion, allq i 's are strictly positive. For convenience, the indication of~ x dependence forq i and ~ v i have been dropped. We have n X i=1 x i E i (~ x) = n X i=1 x i q i I + n X i=1 x i q i ~ v i ~ /I + 1 P n i=1 x i q i n X i=1 x i q i ~ v i ! ~ =I +~ r~ ; (4.24) 26 where ~ r 1 P n i=1 x i q i n X i=1 x i q i ~ v i ! : (4.25) The inverse square root of Eq. (4.24) becomes n X i=1 x i E i (~ x) ! 1 2 / (I +~ r~ ) 1 2 / (Ib~ r~ ); (4.26) where b = 1 p 1j~ rj 2 j~ rj 2 : (4.27) Similarly, we can write n X i=1 (~ x k ) i E i (~ x)/I +~ r k ~ ; (4.28) where ~ r k 1 P n i=1 (~ x k ) i q i n X i=1 (~ x k ) i q i ~ v i ! ; (4.29) and dene~ r k ~ r k ~ r. Inserting Eqs. (4.26) and (4.28) to Eq. (4.22), the condition becomes M y k (~ x)M k (~ x)/ (Ib~ r~ ) (I +~ r k ~ ) (Ib~ r~ ) (4.30) /I + b(~ r~ r k )~ r + ( 1 b 1)~ r k 1~ r~ r k ~ : (4.31) The transformation from Eq. (4.30) to Eq. (4.31) is by the properties of Pauli matrices and 1 2b +b 2 j~ rj 2 = 0 from Eq. (4.27). Hence, M y k (~ x)M k (~ x) = c k (1~ r~ r k )I + b(~ r~ r k )~ r + ( 1 b 1)~ r k ~ ; (4.32) where thec k 's arepositive constants to be determined. The condition to be a measurement, n X k=1 M y k (~ x)M k (~ x) =I; (4.33) 27 requires 8 > > > > < > > > > : n X k=1 c k (1j~ rj 2 ) n X k=1 c k ~ r k ~ r = 1; b ~ r n X k=1 c k ~ r k ! ~ r + 1 b 1 n X k=1 c k ~ r k = 0: (4.34) It is sucient to solve the following conditions, 8 > > > > < > > > > : n X k=1 c k = 1 1j~ rj 2 ; n X k=1 c k ~ r k = 0: (4.35) Note thatj~ rj cannot be 1 becausej~ rj = 1 happens only if theE i 's areall proportional to the same projector, but this contradicts the assumption that the E i 's are linearly independent. The question now becomes whether there exists a set of c k 's (> 0) and ~ x k 's (24 n1 ) such that Eq. (4.35) is satised. First of all, P n i=1 E i (~ x) =I implies that ~ q = (q 1 ; ;q n )24 n1 and n X i=1 q i ~ v i = 0: (4.36) The ~ v i 's and ~ r are 3-dimensional Bloch vectors and ~ x is an n-dimensional vector in4 n1 , wheren can only be 2, 3 or 4 in our case. It is shown in Appendix that there is a one-to-one map between4 n1 and the \allowed" space of~ r, denoted as4 ~ r , 4 ~ r f~ rj~ r = 1 ~ x~ q n X i=1 x i q i ~ v i ! ;8~ x24 n1 g: (4.37) Forn = 2,4 1 is a line and4 ~ r is a line spanned by a pair of parallel and opposite-directioned vectors~ v 1;2 ; for n = 3,4 2 is a regular triangle and4 ~ r is a 2-D triangle stretched by~ v 1;2;3 ; for n = 4,4 3 is a regular tetrahedron and4 ~ r is a tetrahedron stretched by ~ v 1;2;3;4 . An example of the n = 3 case is illustrated in Fig. 4.2. Note that the ~ v i 's are E i (~ x)'s Bloch vectors, which are rotated from the Bloch vectors of the original E i 's. 28 Figure 4.2: The map from4 2 to a triangle4 ~ r in the Bloch sphere. It is clear that there are many solutions to Eq. (4.35); one simple solution is 8 > > < > > : c k = 1 n(1j~ rj 2 ) ; 8k2f1; ;ng; n X k=1 ~ r k = 0; (4.38) where the ~ r k =~ r k ~ r are vectors lying in4 ~ r . For example, when n = 3, the three vectors ~ r k 's lie in a triangle and add up to zero vector. The three ~ r k 's are given by three ~ r k 's, which correspond to three ~ x k 's in4 n1 . In fact, we can solve for the set of ~ x k 's if and only if we can solve for the set of ~ r k 's. 4.2 Constraints by destructive weak measurements So far we have shown that it is possible to perform a weak measurement while staying in a simplex. However, it was assumed that we can choose arbitrary sets of~ x k 's, which correspond to arbitrary weak measurements. We now show that even using a more restricted class of weak measurements|destructive weak measurement|Eq. (4.35) still can be satised. Therefore we can perform a series of destructive weak measurements such that it corresponds to a random walk in a simplex. However, the choice of~ r k 's in Eq. (4.35) are more restricted in this case. The model of destructive weak measurements in [9] consists of coupling the system with an ancilla qubit in statej0i by a weak swap operation, followed by a projective POVM 29 on the ancilla. The outcome ^ P k = s k je k ihe k j of this projective POVM corresponds to a measurement operator M k = p s k 0 @ he k j0ie i ihe k j1i sin 0 he k j0i cos 1 A (4.39) on the system. From the other direction, as shown in [9], for a set of eigenvectorsfjb k i;jb k ig of M y k M k , where jb k i = k j0i + k e i k j1i; jb k i = k j0i k e i k j1i; (4.40) and k ; k 0; 2 k + 2 k = 1; k 2R; (4.41) there exists a corresponding vectorje k i of ^ P k on the ancilla that will give us those eigenvec- tors. With thisje k i, the eigenvalues of M y k M k take the form k ; k = 1 2 s k 1 + p 1 4 2 k 2 k cos 2 sin 2 p 1 4 2 k 2 k cos 2 ! : (4.42) On the other hand, one can check that if P n k=1 M y k M k = I then the corresponding ^ P k 's on the ancilla form a projective POVM. This result can be translated into the Bloch sphere picture for M y k M k : M y k M k = k jb k ihb k j + k jb k ihb k j = k + k 2 I + k k k + k ^ n k ~ ; (4.43) where the unit vector ^ n k is ^ n k = 0 B B B @ 2 k k cos k 2 k k sin k 2 k 2 k 1 C C C A : (4.44) 30 The explicit expression for M y k M k is M y k M k = s k 2 2 + 2j(^ n k ) z j cos 2 p sin 2 + (^ n k ) 2 z cos 2 ! " I + sin 2 j(^ n k ) z j cos 2 + p sin 2 + (^ n k ) 2 z cos 2 ^ n k ~ # : (4.45) Now the question becomes: can we use the destructive weak measurements to perform a series of measurements which keep the evolution of POVM elements in the simplex? More specically, for a given position ~ x in the simplex, can we match Eq. (4.32) with Eq. (4.45), for a set off~ r k g andfc k g that satisfy Eq. (4.35) and a set offs k > 0g andf k ; k g that satisfy Eq. (4.41)? Note that in Eq. (4.45) the vector ^ n k can be any unit vector, and the overall positive constant, s k 2 2 + 2j(^ n k ) z j cos 2 p sin 2 + (^ n k ) 2 z cos 2 ! ; (4.46) can be any positive number because s k can be freely chosen. The only constraint imposed by Eq. (4.45) is that the Bloch vector's length, sin 2 j(^ n k ) z j cos 2 + p sin 2 + (^ n k ) 2 z cos 2 ; (4.47) is determined by the z-component of ^ n k . On the other hand, the requirement that the POVM stay in the simplex means that the measurement operators must satisfy Eq. (4.32), which can be rewritten as M y k (~ x)M k (~ x) =c k (1j~ rj 2 ~ r~ r k ) 2 4 I + j~ r k j (^ r^ r k )^ r + p 1j~ rj 2 (^ r ? ^ r k )^ r ? 1j~ rj 2 ~ r~ r k ~ 3 5 ; (4.48) 31 where ^ r;^ r k are the unit vectors of~ r;~ r k , and ^ r k = (^ r^ r k )^ r + (^ r ? ^ r k )^ r ? ; ^ r ? ^ r = 0: (4.49) The term (^ r ? ^ r k )^ r ? is just the orthogonal component of ^ r k in the direction ^ r ? . (We use hats ^ v to denote unit vectors.) The constraint is that the length of the Bloch vector of Eq. (4.48) must equal Eq. (4.47), j~ r k j p 1j~ rj 2 sin 2 k 1j~ rj 2 j~ rjj~ r k j cos k = sin 2 j(^ n k ) z j cos 2 + p sin 2 + (^ n k ) 2 z cos 2 ; (4.50) where cos k ^ r^ r k and (^ n k ) z = (1 p 1j~ rj 2 )(^ r) z cos k + p 1j~ rj 2 (^ r k ) z p 1j~ rj 2 sin 2 k : (4.51) Note that the right hand side of Eq. (4.50) depends only on the direction of ~ r k (which is ^ r k ), and it is a number between 0 and sin. We can solve Eq. (4.50) by rst determining the direction of ~ r k and then solving for the length of ~ r k . For a given current position ~ r in the simplex4 ~ r , we rst choose the directions of ~ r k 's by pointing from ~ r toward the vertices of4 ~ r ; i.e., for each outcomek, the direction of~ r k is chosen to be parallel to~ v k ~ r. Therefore, cos k = ^ r^ r k = ^ r ~ v k ~ r j~ v k ~ rj : (4.52) Inserting these directions back into Eq. (4.50), the left hand side becomes an increasing function ofj~ r k j: it increases from 0 to 1 asj~ r k j goes from 0 toj~ v k ~ rj. For the directions ^ r k chosen above, we can aways nd lengthsj~ r k j to satisfy Eq. (4.50). For such ~ r k 's, the c k 's can always be chosen to satisfy Eq. (4.35) because the ~ r k 's are coplanar. Therefore, we have a set of ~ r k 's (and hence ~ x k 's) and c k 's that satisfy the constraints of the model of destructive weak measurement and the requirements to perform a weak measurement that corresponds to a random walk in a simplex. 32 4.3 The probability to obtain an outcome So far we have shown that one can perform, by the model of destructive weak measurements, a series of weak measurements which keep the evolution of the corresponding positive oper- ator in a simplex. In this section, we show that, at long times, the walk in the simplex will approach one of the vertices. Each vertex corresponds to an outcome of this series of weak measurements. The probability of obtaining an outcome k from the series of weak mea- surements approaches the probability of obtaining E k if we performed the original POVM, fE i g n i=1 . Note that we only have to consider the case of linearly independent and projective POVMs (LIPPOVM). The following part will assume the POVM is a LIPPOVM. From the form of the measurement operator in Eq. (4.39), it is evident that after per- forming N steps, the measurement operators accumulate as M k N M k 1 / 0 @ e iN 0 cos N 1 A (4.53) =j0ihj + cos N j1ih1j; (4.54) where is a number depending on the string of outcomes k N k 1 and ji 0 @ e iN 1 A : (4.55) The POVM element of this string of outcomes is E k N k 1 =M y k 1 M y k N M k N M k 1 /jihj + cos 2N j1ih1j: (4.56) Thus for large N, the POVM element approaches a projector,jihj, at a rate cos 2N . On the other hand, because we only consider LIPPOVMs, the simplex,4 ~ r , has vertices on the 33 surface of the Bloch sphere. Those vertices represent the LIPPOVM's elements which are proportional to projectors. The one-to-one correspondence between4 ~ r and4 n1 implies that their vertices are mapped to each other. Recall from Eq. (4.15) that when~ x approaches one of the vertices, the walk approaches a projector, which is one of the elements of the LIPPOVM. Therefore, in the large N limit, the random walk on the simplex given by the destructive weak measurements must approach a vertex. After N steps, we denote the position in the simplex as ~ x (i) k N k 1 if it is close to the vertex E i . The index k indicates the outcome of the th weak measurement, and the th measurement depends on all previous outcomes k 1 k 1 . Recall from Eq. (4.15), that if the walk approaches the vertexE 1 , for example, afterN steps, the POVM element becomes E(~ x (1) k N k 1 ) =C(~ x (1) k N k 1 ) " n X i=1 (~ x (1) k N k 1 ) i E i # ; (4.57) where (~ x (1) k N k 1 ) 1 ! 1; (~ x (1) k N k 1 ) i ! 0 for i = 2; ;n; (4.58) andC(~ x (1) k N k 1 ) is a constant depending on the string of outcomes. The probability of obtain- ing the outcome \1" for a state is p 1 X k N k 1 2\1" Tr h E(~ x (1) k N k 1 ) i ; (4.59) where the summation is over all strings of outcomesk N k 1 such that the walk approaches the vertexE 1 . To show thatp 1 indeed approaches the probability given by the usual Born's rule, Tr [E 1 ], it is equivalent to show that ^ p 1 X k N k 1 2\1" E(~ x (1) k N k 1 )!E 1 : (4.60) 34 We rst claim that the sum of the constants over the strings of outcomes, X k N k 1 2\1" C(~ x (1) k N k 1 ); (4.61) is bounded. Indeed, we can nd a unit vectorji such that for alli2f1; ;ng,hjE i ji6= 0. Such aji exists becausefE i g n i=1 is a qubit LIPPOVM. From Eqs. (4.57) and (4.59), we have minfhjE i jig X k N k 1 2\1" C(~ x (1) k N k 1 ) hj^ p 1 jihj n X i=1 ^ p i ji =hjIji = 1: (4.62) Hence, it is bounded by 1=minfhjE i jig. In the large N limit, for each i2f2; ;ng, (~ x (1) k N k 1 ) i ! 0 for all strings of outcomes k N k 1 . Therefore, we have X k N k 1 2\1" C(~ x (1) k N k 1 )(~ x (1) k N k 1 ) i maxf(~ x (1) k N k 1 ) i g X k N k 1 2\1" C(~ x (1) k N k 1 ) maxf(~ x (1) k N k 1 ) i g minfhjE i jig ! 0; (4.63) where maxf(~ x (1) k N k 1 ) i g is the maximum of (~ x (1) k N k 1 ) i among all strings of outcomes. The coecients, P k N k 1 2\1" C(~ x (1) k N k 1 )(~ x (1) k N k 1 ) i , approach zero for i6= 1. For the same reason, other outcomes won't contribute to \1" either. Because P n i=1 ^ p i = I = P n i=1 E i and the fE i g n i=1 are linearly independent, we have X k N k 1 2\1" C(~ x (1) k N k 1 )(~ x (1) k N k 1 ) i ! 1i ; . . . X k N k 1 2\n" C(~ x (n) k N k 1 )(~ x (n) k N k 1 ) i ! ni : (4.64) 35 This veries that p i does approach Tr [E i ], and the POVM element nally evolves close to a vertex E i with probability approaching Tr [E i ]. 36 Chapter 5 Continuous indirect measurements This chapter is based on the work in the paper [11]. In this chapter, we introduce a method to indirectly detect the value of an operator using local two-body interactions and single-qubit continuous measurements. The approach involves applying an interaction Hamiltonian for the system and additional qubits that are being continuously monitored. The information of the system's value for the observable translates into dierent signatures of the monitored qubits, which we can identify. The setup can be applied to quantum codes where we detect errors by measuring the stabilizers. As an example, we apply this detection scheme to the four-qubit Bacon-Shor, which is an error-detecting code that can detect errors by measuring two-local operators. In this chapter, we focus on the measurements of the weight-four stabilizers in the error-detecting Bacon-Shor code. 5.1 Indirect detections Suppose we want to detect the value of a Pauli operatorO with eigenvalues +1 and1 for a system. We design a HamiltonianH = (k=2)(IO)X m coupling the system to an additional monitor qubit m. It is convenient to rewrite the Hamiltonian in terms of projectors, i.e., H =k X m , where is the projector onto the1 eigenspace. The intuition behind this construction is that m will be static when the system is in theO = +1 eigenspace, while Z m will rotate when the system is in theO =1 eigenspace. By measuring Z m , we gain information about which eigenspace the system is in. Therefore, we continuously measureZ m 37 with measurement rate to indirectly measure the value ofO. The measurement outcomes are given by a continuous output current I(t) [44, 23], with dI =hZ m idt + dW 2 p ; (5.1) where dW is a Wiener process with zero mean and variance dt. The whole system evolves according to (t +dt) = A(dI)(t)A y (dI) Tr [A(dI)(t)A y (dI)] ; (5.2) where A(dI) =e iHdt( dI dt Zm) 2 dt : (5.3) This process drives towards one of the eigenspace ofO. To observe this, we expand Eq. (5.2) using Ito's rule [21, 22]: d =i[H;]dt +(Z m Z m )dt + p (Z m +Z m 2 Tr [Z m ])dW: (5.4) The expectation value ofO evolves as dhOi = 2 p (hZ m OihZ m ihOi)dW; (5.5) wherehi Tr [ ] is the expectation value. Here,hOi is a time-dependent stochastic variable. Since dhOi is proportional to the Weiner increment, the evolution ofhOi is a random walk with a time-varying step size. This implies the following two properties: (1) the ensemble average ofhOi remains constant at its initial value. (2) the variance ofhOi tends to increase with time. The rst property can be observed by the fact that E [dhOi] =dE [hOi] = 0 =) E [hOi] =hOi t=0 : (5.6) 38 The change of the variance ofhOi is d E hOi 2 (E [hOi]) 2 = E d(hOi 2 ) = 4E (hZ m OihZ m ihOi) 2 dt 0: (5.7) Eq. (5.7) implies thathOi tends to deviate from its average which remains at its initial value due to Eq. (5.6). However,hOi is bounded between1 and 1. The increase of the variance implies thathOi approaches either +1 or1 at later times. AshOi becomes close to1, we havehZ m OihZ m ihOi, and the step size of the random walk becomes small. Hence,hOi tends to stabilize at1. WhenhOi is either +1 or1, we havehZ m Oi = hZ m ihOi and dhOi = 0 for all later times. ThereforehOi =1 is stable. This shows that when is constantly monitored by Z m , the process drives it towards an eigenspace ofO. The probabilities of approaching theO =1 eigenspaces also match the probabilities of getting the outcomes1 when anO measurement is directly applied to . This is a direct consequence of Eq. (5.6) and the fact thathOi!1 at later times. In fact, after a period whenhOi!1, E [hOi] =P (hOi! +1)P (hOi!1) =hOi t=0 = Tr I +O 2 (0) Tr IO 2 (0) ; (5.8) where Tr IO 2 (0) are the probabilities of getting the results1 when anO measurement is performed on the system. This implies Tr IO 2 (0) = P (hOi!1). These properties validate the whole process as a properO measurement on the quantum system. 5.1.1 Detection methods The value ofhOi, however, is not directly accessible because only Z m is being continuously measured. In order to learn the value ofO, we can use an estimator ^ , prepared maximally mixed, to evolve according to Eq. (5.2) with the outcomes dI from theZ m measurements of 39 . The information contained in dI steers ^ to the correct eigenspace is in. The following explains this behavior. Since H commutes withO, the evolution from Eq. (5.2) does not cause transitions between the eigenspaces ofO. If a state starts in a block diagonal form, i.e., (0) =p + (0) + (0) +p (0) (0); (5.9) such that Tr [O (0)] =1, p (0) 0 and p + (0) +p (0) = 1, then the state maintains the same block diagonal structure at all later times: (t) =p + (t) + (t) +p (t) (t); (5.10) where Tr [O (t)] =1, p (t) 0 and p + (t) +p (t) = 1 for all t 0. ( (t) are proper density matrices.) To evaluate how p evolves with time, we look at the expectation values of the eigenspace projectors, p (t +dt) = Tr [(t +dt) ] = 1 N p (t) Tr A(dI) (t)A y (dI) ; (5.11) whereN = Tr A(dI)(t)A y (dI) . For innitesimal dt, one can deduce that p (t +dt) 1 N p (t) Tr h (t)e 2( dI dt Zm) 2 dt i 1 N p (t)e 2( dI dt hZmi ) 2 dt ; (5.12) 40 wherehZ m i = Tr [Z m (t)]. (The derivation is in Appendix) This form is essentially the same as Bayes's theorem|our knowledge of the probability of1 given the outcome dI is P (1jdI) = P (dIj 1)P (1) P (dIj + 1)P (+1) +P (dIj 1)P (1) = 1 N e 2 dI(1)dt p dt 2 P (1); (5.13) where the exponential represents the Gaussian distribution of the stochastic variable dI, given the value +1 or1. The evolution for (t) is (t +dt) = (t +dt) Tr [ (t +dt) ] = A(dI) (t)A y (dI) Tr [A(dI)(t)A y (dI)] Tr A(dI)(t)A y (dI) Tr [A(dI) (t)A y (dI)] = A(dI) (t)A y (dI) Tr [A(dI) (t)A y (dI)] ; (5.14) which has the same form as Eq. (5.2). This implies the following fact. If two initial states, 1;2 (0) = 1;2 j0i m h0j, are both in the +1 or both in the1 eigenspace, i.e., hOi 1;2 (0) = +1 orhOi 1;2 (0) =1, and they both evolve according to Eq. (5.2) with the same set of measurement operatorsA(dI), then we havehZ m i 1 =hZ m i 2 for any time. This is because for any state strictly in eitherO =1 eigenspace, the evolution of the whole system only involves the evolution of the monitor qubitm. Since bothm's of 1;2 are initially prepared inj0i m h0j, it is true thathZ m i 1 =hZ m i 2 for all times. These results are sucient to show the steering eect of the estimator. Let us use an estimator ^ to represent our knowledge of a real system real that is constantly monitored through the measurements of Z m . We initialize the estimator to be 41 maximally mixed in the system part to represent our ignorance of the system. The initial state of the estimator can be written as ^ (0) = I d d j0i m h0j = 1 2 2 d + j0i m h0j + 1 2 2 d j0i m h0j =p + (0) + (0) +p (0) (0); (5.15) where d is the system dimension excluding m. The estimator initially has p (0) = 1=2 and (0) = (2=d) j0i m h0j. Suppose the real system real (0) is in theO =1 eigenspace and is coupled to a monitor qubit m prepared in statej0i m h0j. Continuously measuring Z m gives outcomes dI =hZ m i real dt + dW 2 p : (5.16) We use the signal dI from real to evolve ^ according to Eq. (5.2). Since both real (t) and (t) are in theO =1 eigenspace and have the same initial state of m, we have hZ m i real =hZ m i (5.17) for any time t 0. The ratio of the p becomes p + (t +dt) p (t +dt) = p + (t)e 2 hZmi real hZmi + + dW 2 p 2 dt p (t)e 2 dW 2 p 2 dt on average ! p + (t) p (t) e 2(hZmi real hZmi + ) 2 dt : (5.18) It shows that the ratio ofp + =p decreases on average due to the dierence betweenhZ m i real andhZ m i + . SinceH =k X m , only the negative eigenspace causes transitions. Therefore, it is evident thathZ m i real 6=hZ m i + = 1 for most times. It is expected that p ! 1 and p + ! 0 at later times. 42 If real is in theO = +1 eigenspace, the ratio becomes p + (t +dt) p (t +dt) = p + (t) p (t) e 2(hZmi real hZmi ) 2 dt ; (5.19) wherehZ m i real =hZ m i + = 1. We will have p + ! 1 and p ! 0 instead. The above shows that when is in an eigenspace ofO, the measurement records dI drive ^ to that eigenspace. IfhOi ^ approaches +1, we learn that real is in the eigenspace ofO = +1. IfhOi ^ approaches1 then real is in the eigenspace ofO =1. These results are sucient for error detections on general stabilizer codes, where we prepare the encoded state in the joint +1 eigenspace of a set of commuting operators. For each stabilizerO i , we attach an extra qubitm i to the system with Hamiltonian (1=2)(IO i )X m i and continuously measure Z m i . From the signals of measuring Z m i , we are able to detect if errors have taken the state out of the stabilized space. However, simulating the evolution of the estimator requires computational overhead. As the system size grows, the exponential increase of the matrix dimension makes the method of simulating the estimator impractical. We provide in the following an alternative method to retrieve the information contained in the outcomes dI without simulating the whole quantum state. Note that in this particular setup where H =k X m , it is clear that if the state is in the +1 eigenspace thenhZ m i = 1 at all times. The signal becomes a Wiener process with a constant drift, i.e.,dI = 1dt+(dW=2 p ). We can evaluate an average function ofdI dened by I(t) 8 > > < > > : 1 t R t 0 dI if 0tw 1 w R t tw dI if w<t ; (5.20) where w is the window width which is short compared to the average time between errors (1/the rate of errors) but long compared to the inverse of the measurement rate on the 43 monitor qubits (1=), i.e., (1=)w (1/the rate of errors). In the case where is in the +1 eigenspace, the average function reads I(t) = 1 + 8 > > < > > : 1 t R t 0 dW 2 p if 0tw 1 w R t tw dW 2 p if w<t : (5.21) The variance of I(t) is Var I(t) = 8 > > < > > : E 1 t R t 0 dW 2 p 2 = 1 4t if 0tw E 1 w R t tw dW 2 p 2 = 1 4w if w<t : (5.22) Because E I(t) = 1 and Var I(t) is inversely proportional to time, we should expect that I(t) converges to 1 after t w. If is in the1 eigenspace, there will be oscillations of hZ m i. The dynamics ofhZ m i involve dhZ m i = 2khY m idt + 2 p (1hZ m i 2 )dW; (5.23) dhY m i =2khZ m idt 2hY m idt 2 p hZ m ihY m idW: (5.24) The X m in H causes a rotation of the y-z plane in the Bloch sphere for the monitor qubit m. The rst terms in above equations indicate such a rotation. The exponential suppression in the second term forhY m i is due to the measurements on Z m . The average function of dI becomes I(t) =hZ m i + 8 > > < > > : 1 t R t 0 dW 2 p if 0tw 1 w R t tw dW 2 p if w<t ; (5.25) wherehZ m i denotes the average value ofhZ m i over an integration period, i.e., hZ m i = 8 > > < > > : 1 t R t 0 hZ m idt if 0tw 1 w R t tw hZ m idt if w<t : (5.26) 44 Since there are oscillations ofhZ m i between1 to 1,hZ m i should be noticeably smaller than 1. The later simulation shows that I(t) approaches zero after a period of time, when is in the1 eigenspace. By directly evaluatingI(t) from the measurement outcomes, one can determine whether the state is in the +1 eigenspace. Although this method is noisier than the method of calculatinghOi ^ from the estimator, it signicantly speeds up the process of detecting errors. The relative size between the strength of the Hamiltonian k and the measurement rate plays a role in determining the eectiveness of this indirect detection scheme. If is too large, then the frequent measurements on Z m freeze m in the statej0i m h0j due to the quantum Zeno eect. In this case,hZ m i stays close to 1 for a much longer time, and the information gain is greatly reduced. If is too small, the ratio, in Eqs. (5.18) and (5.19), between p + and p changes slowly. The rate at which the estimator approaches either1 eigenspace becomes small. This is also not an ideal limit for learning the value ofO for . From our testing, the most ecient regime is around = 0:5k 1:5k. In most cases, the stabilizers are high weight operators, e.g., weight four stabilizers in the surface code. Directly measuring these high weight operators requires multiple gate operations, which can be more inaccurate. This passive indirect detection scheme can pro- vide an alternative way to measure these stabilizers. In Sec. 5.3, we show how the desired Hamiltonians can be eectively constructed by 2-local operators. In the following subsection, we provide a minimal example demonstrating the process and the behavior of the indirect detection method. We set = 0:6k and the time unit to be 1=k throughout the rest of the paper. 5.1.2 ZZ example We provide a simple 3-qubit example to demonstrate the indirect detection scheme. Suppose we want to know the value of the operatorZ 1 Z 2 for qubits 1 and 2. We bring in an additional monitor qubit m and turn on the joint Hamiltonian H = (k=2)(I Z 1 Z 2 )X m . Under continuous measurements of Z m with outcomes dI, the whole state evolves according to 45 Figure 5.1: (a), (b), (e) and (f) are the estimator approach. The signals from measuring the physical state drive the estimator to the Z 1 Z 2 =1 eigenspace the physical state is in. (a) and (e) are the evolutions ofhZ 1 Z 2 i ^ when is in the +1 eigenspace of Z 1 Z 2 . (b) and (f) are the cases when is in the1 eigenspace of Z 1 Z 2 . (c), (d), (g) and (h) represent the average function I(t). It converges to 1 when the physical state is in the +1 eigenspace, and it converges to 0 when the state is in the1 eigenspace. Eq. (5.2). In experiment,dI are obtained from the measurement apparatus. For simulation, the outcomes are generated using dI = Tr [Z m ]dt +dW=(2 p ), where dW is a Wiener process. Figs. 5.1a and 5.1b show the dragging eect of the estimator|if the state is in the Z 1 Z 2 =1 eigenspace then the estimator ^ approaches the eigenspace is in. From the value thathZ 1 Z 2 i ^ approaches, one can know which eigenspace is in. The plots Figs. 5.1e and 5.1f are the ensemble average over 500 trajectories ofhZ 1 Z 2 i ^ . The other method to translate the information contained indI is to evaluate the average functionI(t) dened by Eq. (5.20). For convenience, we choosew = 40=k and evaluateI(t) from time 0 tow. Figs. 5.1c and 5.1d illustrates the dierence between the state being in the1 eigenspaces. I(t) converges to 1 if is in the +1 eigenspace, and it converges to 0 if is in the1 eigenspace. In these example, the initial state of is(0) = (1=2)(j00i+j11i)(h00j+h11j) j0i m h0j for the case of Z 1 Z 2 = +1 and is(0) = (1=2)(j01i +j10i)(h01j +h10j) j0i m h0j for the case ofZ 1 Z 2 =1. 46 Note that there is a trade-o between accuracy and eciency for the two methods|the estimator approach gives a more stable readout comparing to I(t) but requires computational overhead. The estimator approach can be more accurate for theoretical analysis while the average function is more experimentally feasible. 5.2 An application to the four-qubit Bacon-Shor code The 4-qubit Bacon-Shor code is an error-detecting code that can detect errors by measuring only weight-two operators. In the stabilizer formalism, it has two weight-four stabilizers, S z = ZZZZ and S x = XXXX. Checking if the system stays in the joint S z = +1 and S x = +1 eigenspace allows us to detect single-qubit errors. To measure the stabilizers, we could in principle bring in two extra qubits m z and m x and apply the Hamiltonian H = k 2 (IZ 1 Z 2 Z 3 Z 4 )X mz + k 2 (IX 1 X 2 X 3 X 4 )X mx ; with continuous measurements on Z mz and Z mx . However, applying the weight-ve Hamil- tonian requires many-body interactions and is experimentally hard. As we show in Sec. 5.3, the above Hamiltonian would appear in the fth-order expansion of the perturbation con- struction. It means that the base Hamiltonian should be ve orders of magnitude stronger than the Hamiltonian needed for indirect detection. This poses a practical challenge for experiments. To reduce the energy scale, we can instead use H = k 2 (Z 1 Z 2 Z 3 Z 4 )X mz + k 2 (X 1 X 3 X 2 X 4 )X mx ; (5.27) which involves only 3-local interactions. As shown in Sec. 5.3, this Hamiltonian appears in the second order expansion of the perturbative construction, where m z andm x are eective two-level systems. To gain insight into this setup, let us rst recall the stabilizer formalism for the 4-qubit Bacon-Shor code. The code uses four physical qubits to encode one logical qubit 47 j 0 0 0 0i = 1 p 2 (j0000i +j1111i), j 0 1 0 0i = 1 p 2 (j0101i +j1010i), j 0 0 1 0i = 1 p 2 (j0000ij1111i), j 0 1 1 0i = 1 p 2 (j1010ij0101i), j 1 0 0 0i = 1 p 2 (j0011i +j1100i), j 1 1 0 0i = 1 p 2 (j1001i +j0110i), j 1 0 1 0i = 1 p 2 (j1100ij0011i), j 1 1 1 0i = 1 p 2 (j0110ij1001i), j 0 0 0 1i = 1 p 2 (j0001i +j1110i), j 0 1 0 1i = 1 p 2 (j0100i +j1011i), j 0 0 1 1i = 1 p 2 (j1110ij0001i), j 0 1 1 1i = 1 p 2 (j0100ij1011i), j 1 0 0 1i = 1 p 2 (j1101i +j0010i), j 1 1 0 1i = 1 p 2 (j1000i +j0111i), j 1 0 1 1i = 1 p 2 (j0010ij1101i), j 1 1 1 1i = 1 p 2 (j1000ij0111i). Table 5.1: Code basis: bar/un-bar represents encoded/physical basis. and can detect any single-qubit error. The Hilbert space decomposes into tensor products of four subsystems with the following set of commuting operators and their complements, Logical qubit : Z L =Z 1 Z 3 X L =X 1 X 2 Gauge qubit : Z G =Z 1 Z 2 X G =X 1 X 3 Stabilizers : S x =X 1 X 2 X 3 X 4 S x =Z 4 S z =Z 1 Z 2 Z 3 Z 4 S z =X 1 X 2 X 3 : The encoded basis isjZ L Z G S x S z i. We add a bar on each bit for the encoded basis to distinguish it from the physical basis. For example,j 0 1 0 1i represents the basis vector corresponding to Z L = +1, Z G =1, S x = +1 and S z =1. The relationship between the two bases can be found in Table 5.1. In this encoded basis, it is convenient to rewrite Eq. (5.27) as H = k 2 (IS z )Z G X mz + k 2 (IS x )X G X mx =k Sz Z G X mz +k Sx X G X mx ; (5.28) where Sz and Sx are projectors onto their1 eigenspaces. The signature for the state being in either combination of S z =1 and S x =1 is clear: when the state is in the S z = +1 andS x = +1 eigenspace, the monitor qubits are static; when either S z orS x is1, there are oscillations for Z mz or Z mx . Note that since there is no term involving Z L or X L , 48 the logical qubit is perfectly preserved during the process of indirect detection. The gauge qubit can be treated as an external degree of freedom for the system, where its dynamics are irrelevant. The non-commutativity between the two terms inH is on the gauge system, and does not aect the detection process. We prepare the state in the simultaneous +1 eigenspace ofS z andS x and continuously monitorZ mz andZ mx . If there is no error, we should observe static values ofhZ mz;x i, which are both one in our setting. If an error takes the state out of the +1 eigenspace of a stabilizer then we can detect it by the non-static evolution of hZ mz;x i. However, these expectation values are not directly obtained from experiments. The outcomes of the continuous measurements are dI z;x =hZ mz;x idt +dW z;x =2 p . To retrieve information contained inhZ mz;x i, we can evaluate the time average of the signals dened in Eq. (5.20). We can also use an estimator ^ to learn the stabilizer values as described in Sec. 5.1. From the outcomes dI z;x =hZ mz;x i dt +dW z;x =2 p , we evolve the estimator according to ^ (t +dt) = A^ (t)A y Tr [A^ (t)A y ] ; (5.29) where A =e iHdt( dIz dt Zmz) 2 dt( dIx dt Zmx) 2 dt : (5.30) The estimator is initially maximally mixed and can be decomposed into four blocks, i.e., ^ (t) = X =1;=1 p (t) (t); (5.31) where p (t) = Tr Sx Sz ^ (t) and (t) = Sx Sz ^ (t) Sz Sx . The evolutions for the probabilities become p (t +dt) (5.32) p (t) N e 2 h ( dIz dt hZmz i ) 2 ( dIx dt hZmx i ) 2 i dt : 49 Figure 5.2: I x;z (t) for the four eigenspaces. The red is I z (t) and the blue is I x (t). The average function converges to 1 (or 0) if the corresponding stabilizer is in the +1 (or1) eigenspace. Figure 5.3: The estimator approach. The blue is the evolution ofhS x i ^ . The red is the evolution ofhS z i ^ . It is shown that the estimator approaches the eigenspace the system belongs to. When is in the eigenspace of S x = and S z =, the p of the estimator has the largest increase on average. Hence, the estimator approaches the eigenspace of S x = and S z =. The argument mostly follows the discussion in Sec. 5.1 for each S x and S z . Simulations of the time-averaged signal and of the estimator approach, over 500 trajec- tories, are shown in Fig. 5.2 and Fig. 5.3. We use the following initial states as examples for the system being in the four eigenspaces of S x =1 and S z =1: ++ 1 2 (j 0i +j 1i)(h 0j +h 1j) j 0 0 0ih 0 0 0j; + 1 2 (j 0i +j 1i)(h 0j +h 1j) j 0 0 1ih 0 0 1j; + 1 2 (j 0i +j 1i)(h 0j +h 1j) j 0 1 0ih 0 1 0j; 1 2 (j 0i +j 1i)(h 0j +h 1j) j 0 1 1ih 0 1 1j; (5.33) 50 Figure 5.4: An X 1 error happened at t = 20 as indicated by the black line. After the error, I z starts to approach 0 andhS z i ^ ips to1. where they are expressed in the encoded basisjZ L Z G S x S z i. (0) is one of the states above with monitor qubits initialized in statej0ih0j. 5.2.1 Error detection and suppression When we apply the four-qubit Bacon-Shor code, we prepare the state in the S x = +1 and S z = +1 eigenspace and store information in the logical qubit of the state. To detect errors, we attach monitor qubits to the system and continuously measure them. If there are no errors, the monitor qubits are static and both I z;x (t) converge to 1. Or we can simulate the estimator, which will approach the joint +1 eigenspace. Let us rst consider single-qubit errors. Suppose an X 1 error happened on the rst system qubit. The error anticommutes with S z and the state is taken to the S z =1 eigenspace. A sample trajectory is shown in Fig. 5.4, where the error is detected by observing that I z (t) drifts to 0 andhS z i ^ ips to1. We present another example where the errors are continuous-in-time 1/f Hamiltonian errors, i.e., H err (t) = X i i (t) i ; (5.34) 51 Figure 5.5: (a) showshS z;x i for the system. After a period, S x is ipped to1 while S z remains at +1. (b) shows I z;x (t). After a period, I x (t) starts to decay while I z (t) remains at +1. (c) showshS z;x i ^ for the estimator.hS x i ^ approaches +1 initially, but it ips to1 afterhS x i!1. where each i is a single-qubit Pauli matrix acting on the ith qubit and i (t) is a time- dependent scalar function. Each i (t) consists of exponentially decaying random pulses with magnitude , i.e., i (t) = X i (tt i ) exp tt i ; (5.35) where (t) is the Heaviside step function [30]. It has been shown that this type of error can be suppressed by implementing continuous indirect measurements. For most trajectories, the system stays close to the S z;x = +1 eigenspace. I z;x (t) converges to 1 whilehS z;x i ^ approaches 1 and stays at 1. Occasionally, the error can cause the system to jump to the 1 eigenspace ofS z;x . A sample trajectory of this case is shown in Fig. 5.5, where we detect the system'shS x i jumping to1 by observing that I x (t) starts to decay to 0 andhS x i ^ ips to1. In general, since any single-qubit Pauli error anticommutes with at least one of the S z;x , any single-qubit error can be detected. Multi-qubit errors can be detected when they consist of operators anticommuting with one of the S x;z . The undetectable errors are those commuting with both S x;z . However, they must be at least weight 2. They happen at lower rates than single-qubit errors. 52 Figure 5.6: A sample of I x;z (t) with anX mz error att = 20 and anX 1 error att = 100. The red curve represents I z (t), and the blue curve represents I x (t). I x (t) remains at 1 because the errors commute with S x . I z (t) ips to1 due to the X mz error and then converges to 0 after the X 1 error happened. The window width w is set to 40=k in this example. We now consider the cases when errors happen on the monitor qubits. Note that the essential indicator that allows us to distinguish the four eigenspaces of S x;z =1 is whether hZ mz;x i is static or oscillatory in motion. WhenS x;z is +1,hZ mx;z i is static. WhenS x;z is1, hZ mx;z i is oscillatory. It turns out that the process of stabilizer detection can be preserved with low rate errors on the monitor qubits. Let us begin with the case of instantaneous errors on the monitor qubits. Suppose an X mz error happened on the monitor m z . IfhZ mz i was static (because the system is in theS z = +1 eigenspace),hZ mz i ips from +1 to1 but remains static. As shown in Fig. 5.6, I z (t) converges to1 after an X mz error happened at t = 20, and then a subsequent X 1 error happened at t = 100 is detected by observing I z (t) evolving to 0. IfhZ mz i is oscillatory, an X mz ips the value ofhZ mz i but does not change the oscillatory motion. In general, I x;z (t) converging to1 indicates that there was no error, and either I x;z (t) converging to 0 indicates there was an error. When such instantaneous errors happen with rates much smaller than 0:1, which is approximately the inverse of the measurement time for the indirect detection, the error detection process is preserved. For continuous Hamiltonian errors, since the monitors are being continuously measured, errors with strength much smaller than the measurement rate are partially suppressed by the quantum Zeno eect. Hence, the process of detecting errors on the encoded four- qubit system can be preserved even with low-rate errors on the monitors. However, if the 53 Figure 5.7: The evolutions of the stabilizers under various error models. (a) is the case for 1/f Hamiltonian noise. (b) is the case for constant Hamiltonian errors. (c) is the case for white noise. They are ensemble averages over 1000 trajectories. errors happen at high rates, it can cause rapid ipping that mimics the oscillatory eect of Z mz , which normally would occur only when S z =1. In that limit indirect detection becomes ineective. Of course, these conclusions are for the particular error model we have been considering. Experimental measurements of the error process might suggest alternative versions of this scheme, e.g., measuring X mz instead of Z mz if there are mainly X mz errors. In the full construction described in Sec. 5.3, the qubits m z and m x are two pairs of physical qubits (a;b) and (c;d). Each pair (a;b) and (c;d) is conned to the ground space of the strong base Hamiltonian, (K=2)(IZ a Z b ) + (K=2)(IZ c Z d ). X a X b and X c X d in the eective Hamiltonian cause transitions only within the ground space, i.e.,j00i$j11i. Hence they act as X mz;x for the eective two qubits m z and m x . The process of detecting errors for the encoded system is similar: when S z = +1, both a and b are static; when S z =1, both a and b are oscillatory. The same applies to c and d for S x . We only need to continuously measure one qubit for each pair, e.g., measuring a and c. If errors can happen on the monitor qubits, it follows similarly from the above argument that the error detection process for the system qubit can be preserved under low-rate errors. It is well-known that frequent measurements can freeze a system in an eigenspace of the measurement observable due to the quantum Zeno eect. There have been many eorts 54 to harness the Zeno eect for error suppression [36, 45, 26]. In [34], it is shown that non- Markovian errors can be suppressed by the quantum Zeno eect while Markovian errors can not. In this subsection, we investigate error suppression for various models under continuous indirect measurements. We rst consider the 1/f Hamiltonian errors dened in Eq. (5.34), where the sum is over all physical qubits. In Fig. 5.7, we plot the ensemble average of the system's stabilizer values under this 1/f Hamiltonian error. As shown in Fig. 5.7a, the red and blue curves represent the case with indirect detection while the purple and yellow curves represent the case without the measurement setup. The pulse rate and are 0:1k and 1=k, wherek is the strength of the Hamiltonian. The measurement rate is set to 0:6k. The red and blue curves decay noticeably more slowly than the purple and yellow, which shows that the system state tends to remain in the S z;x = +1 eigenspace in the presence of indirect stabilizer detection. Note that 1/f Hamiltonian noise is a type of non-Markovian error process. The exhibited suppression aligns with the result in [34] that non-Markovian errors can be suppressed by the quantum Zeno eect. We present another example of non- Markovian errors, where the errors are constant Hamiltonian terms, i.e., H err = P i . To keep the same error magnitude as in the 1/f noise case, the is set to 0:01k, which is the average strength of i (t) in the 1/f noise. The result is shown in Fig. 5.7b. Convergence to the joint +1 eigenspace of S z;x is apparent when indirect detection is applied. Another method to benchmark the state protection is to evaluate the trace distance between the state at time t and the initial state [36]. The smaller the trace distance the closer the state remains to its initial state. Fig. 5.8 shows a clear protection of the state when the system is being measured. However, when the errors are Markovian (white noise) the measurements do not appear to x the stabilizer values, as shown in Fig. 5.7c. For Markovian noise, the probability of a state transition is of order dt for a time step. Errors of this type cannot be suppressed by frequent measurements and full error correction is required to protect the states. For non-Markovian noise, by contrast, the probability of a state transition is of order dt 2 in a 55 Figure 5.8:D(t) = 1 2 jj(t)(0)jj 1 . The red curve is the case without measurements while the blue curve is the case with continuous indirect detection. They are ensemble averages over 1000 trajectories. time step. This is why these transitions can be suppressed by the quantum Zeno eect [34]. The above simulations for continuous indirect measurements agree with these results. It is worth noting that for the purpose of error prevention, it is possible in principle to suppress Hamiltonian errors by applying a strong Hamiltonian alone. For example, suppose we have a qubit prepared in thej0i state in theZ basis with the presence of anX Hamiltonian error. The error can cause the state to rotate on the y-z plane in the Bloch sphere. However, if we apply a Z Hamiltonian, which is strong comparing to the error term X, the rotating axis becomes closely aligned with the z-axis. The evolution for the state will be conned in a small region near the north pole. The region can be made smaller as we increase the strength for theZ term. Therefore, the state is maintained close to its initial statej0i. Recall the setup in the indirect measurements. We require interaction Hamiltonians between the system and the monitor qubits. These Hamiltonians also contribute to the suppression of errors because of the above axis-pinning behavior. However, if an error term has a time- dependent coecient with a frequency component on resonance with energy dierences in the system Hamiltonian, transitions are not suppressed. In this case, applying the Hamiltonian alone is not eective against the error. However, applying both the Hamiltonian and the continuous measurements on the monitor qubits performs better in these cases as shown 56 Figure 5.9: The evolution ofhS z i under an on-resonance X 1 error. The blue curve is the case when the full indirect detection scheme is applied. The red curve is the case when the Hamiltonian for indirect detection is applied but no measurements are made on the monitor qubits. in Fig. 5.9. Overall, continuous indirect measurements can protect encoded states against errors. 5.3 Constructing the Hamiltonian for indirect detec- tion In this section, we show how to build an eective Hamiltonian for indirect detection. The method is based on the idea of perturbation gadgets [25]. It uses 2-local Hamiltonians to produce an eective k-local Hamiltonian that appears in the rst non-vanishing order for the low-lying energy eigenstates. We begin by brie y recapping the theory presented in [25]. Suppose we have a strong base HamiltonianH (0) and a weak potentialV . H (0) has zero ground state energy with a degenerate ground spaceG (0) spanned by eigenvectorsje 0 1 i:::je 0 d i, andV weakly perturbs it. The total HamiltonianH =H (0) +V will have ad-dimensional vector spaceG spanned by the d lowest energy eigenstatesje 1 i:::je d i. For small enough , G largely overlaps withG 0 . The space spanned by the lowest d energy eigenstates has an eective Hamiltonian H e d X i=1 E i je i ihe i j; (5.36) 57 which can be expanded in powers of , i.e., H e =U 0 @ 1 X m=1 m X (m1) P 0 VS l 1 VS l m1 VP 0 1 A U y : (5.37) The operatorP 0 projects any vector onto the unperturbed ground spaceG (0) , and the linear operatorU satises UP 0 je i i =je i i and UG (0)? = 0: (5.38) The operator S l is S l = 8 > > > < > > > : X i>0 P i (E (0) i ) l if l> 0; P 0 if l = 0; (5.39) whereP i is the projector corresponding to the energy levelE (0) i of the base HamiltonianH (0) . The summation P (m1) is over nonnegative integersl 1 ;l 2 ;:::;l m1 such thatl 1 ++l m1 = m 1 and l 1 + +l x x for any x from 1 to m 1. U andU y can also be expanded in powers of but only their zeroth order terms, which are bothP 0 , will contribute in the later discussion. A more detailed derivation of these results can be found in [25]. Note that the expansion converges only ifjjVjj < E (0) =4, where E (0) is the energy gap between the ground energy (assumed zero) and the second lowest energy. To have a good approximation from the perturbation, we would expectjjVjj to be much smaller than E (0) . In this limit, the eect of adding V to H (0) becomes a small splitting of the degenerate ground space with a small deviation from the ground spaceG (0) toG. When an initial state is prepared inG (0) , its evolution stays mainly inG and the eective Hamiltonian H e will be a good approximation for H. The following construction for the indirect measurement requires us to design 2-local Hamiltonian termsH (0) andV such that the rst non-vanishing order of the expansion gives the desired Hamiltonian. 58 5.3.1 First example: ZZ detection Suppose we want to measure Z 1 Z 2 for qubits 1 and 2, and the desired Hamiltonian is H target = k 2 (IZ 1 Z 2 )X m : We bring in two ancillary qubits m 1 and m 2 , and turn on a Hamiltonian H = H (0) +V , where H (0) = K 2 (IZ m 1 Z m 2 ) (5.40) and the perturbing term is V =K (Z 1 X m 1 +Z 2 X m 2 + 2X m 1 X m 2 ): (5.41) K is a constant and 1. (Note that the identity term in the base Hamiltonian is unnecessary but we keep it for convenience.) The expansion of H e in Eq. (5.37) up to second order in gives H e =U P 0 VP 0 + 2 P 0 VS 1 VP 0 +O( 3 ) U y =K 2 P 0 [2 (IZ 1 Z 2 )X m 1 X m 2 2]P 0 +O( 3 ): (5.42) The ancillary qubits are prepared in the ground space,G (0) =H 12 spanfj00i;j11ig m 1 m 2 , and the eective Hamiltonian for the 4-qubit system can be approximated by ~ H e = 2K 2 (IZ 1 Z 2 )X m 1 X m 2 ; (5.43) in the limit of 1. The shifted term proportional to P 0 is neglected since it acts as the identity in the subspace. Note that since the ancillary qubits are restricted toG (0) , which is a 2-dimensional subspace, we can treat m 1 andm 2 as an eective qubitm and the operator 59 X m 1 X m 2 behaves as X m that ips m. Hence, it can be simplied as a 3-body system with Hamiltonian ~ H e = 2K 2 (IZ 1 Z 2 )X m ; (5.44) which is in the desired form for the indirect measurement (with k = 4K 2 ). Since m 1 and m 2 are conned to the ground spaceG (0) and are simultaneously rotated by X m 1 X m 2 , we can detect the value of Z 1 Z 2 by continuously measure only one of Z m 1 or Z m 2 . When the state is in the eigenspace of Z 1 Z 2 = +1, both m 1 and m 2 are static. When the state is in the Z 1 Z 2 =1 eigenspace,hZ m 1 i andhZ m 2 i are oscillatory. The system's Z 1 Z 2 value can be obtained by calculating the estimator or evaluating the time average of the signal as described above. 5.3.2 Construction for the four-qubit Bacon-Shor code To indirectly measure the stabilizers, S z =Z 1 Z 2 Z 3 Z 4 and S x =X 1 X 2 X 3 X 4 , for the 4-qubit Bacon-Shor code, we apply the Hamiltonian, H = k 2 (Z 1 Z 2 Z 3 Z 4 )X mz + k 2 (X 1 X 3 X 2 X 4 )X mx ; (5.45) and continuously measure Z mz and Z mx . However, to obtain this Hamiltonian using only 2-local operators requires four ancillary qubits, which we call a;b;c andd. The full physical system becomes an 8-qubit state, where 1; 2; 3; 4 are the system qubits and a;b;c;d are the monitor qubits for the indirect measurements. The full perturbative construction has a Hamiltonian H =H (0) +V , where the base Hamiltonian is H (0) = K 2 (IZ a Z b ) + K 2 (IZ c Z d ); (5.46) 60 and the perturbing term is V = K 2 p 2 (Z 3 +Z 4 Z 1 Z 2 )X a + K 2 p 2 (Z 3 +Z 4 +Z 1 +Z 2 )X b + K 2 p 2 (X 2 +X 4 X 1 X 3 )X c + K 2 p 2 (X 2 +X 4 +X 1 +X 3 )X d + K 2 (Z 1 Z 2 +Z 3 Z 4 +X 1 X 3 +X 2 X 4 ): (5.47) The monitor qubits are prepared in the ground space of H (0) , which consists of two two-level subspaces for the monitors. The unperturbed ground space is G (0) = H sys spanfj00i;j11ig ab spanfj00i;j11ig cd . After adding V , the perturbed ground space has an eective Hamiltonian that reads H e =P 0 VP 0 + 2 P 0 VS 1 VP 0 +O( 3 ) =K 2 1 2 (Z 1 Z 2 Z 3 Z 4 )X a X b P 0 (5.48) +K 2 1 2 (X 1 X 3 X 2 X 4 )X c X d P 0 K 2 2P 0 +O( 3 ): Since the ancillary qubits are prepared in the ground space ofH (0) , the full system eectively has the Hamiltonian ~ H e = K 2 2 [(Z 1 Z 2 Z 3 Z 4 )X a X b + (X 1 X 3 X 2 X 4 )X c X d ]: (5.49) The X a X b and X c X d only cause transitions within the ground spaceG (0) , and they act as a single-qubit X for an eective qubit conned in the space spanned byfj00i;j11ig. We obtain the target Hamiltonian (5.45) by identifying X a X b ! X mz and X c X d ! X mx . The monitors are initially prepared inj0000i abcd . X a X b (X c X d ) simultaneously rotates Z a and 61 Figure 5.10: The evolution ofhS z i under the full 8-qubit construction. The orange curve includes the indirect detection alone without any errors. The blue curve also includes 1/f Hamiltonian noise. The green curve is the 1/f Hamiltonian noise alone without any indirect measurement. They are ensemble averages over 500 trajectories. Z b (Z c and Z d ) when the state is in the S z =1 (S x =1) eigenspace. To measure the values of S z and S x , we continuously measure Z a (or Z b ) and Z c (or Z d ). The information of the system being in either eigenspace of S z;x =1 can be obtained by evaluating I a (t) and I c (t) or by calculatinghS z;x i ^ using an estimator ^ as described in Sec. 5.1. When the system is in the S z = +1 eigenspace,hZ a i is static. I a (t) converges to +1 andhS z i ^ ! +1. When the system is in the S z =1 eigenspace,hZ a i is oscillatory. I a (t) approaches 0 and hS z i ^ !1. The same detection rule applies to S x . It is worth recalling that the constant K 2 in the eective Hamiltonian is the strength k of the target Hamiltonian in Eq. (5.45). The fact that needs to be small for the per- turbation to work accurately implies that K, the strength of the base Hamiltonian, has to be large enough that K 2 is large compared to the error strength (or rate). We numerically simulated an example with 2 0:001 to demonstrate the performance of the full perturba- tive construction. The result forhS z i is shown in Fig. 5.10. (hS x i behaves similarly.) The ensemble averages of trajectories forhS z i are plotted for various cases. The orange curve represents the no-error case when we apply the full construction using only 2-local Hamilto- nians from Eqs. (5.46) and (5.47) and continuous measurements of Z a and Z c . When there is no error,hS z i is expected to remain 1 throughout the detection process. This is true for the 6-qubit setup introduced in Sec. 5.2. However, building the Hamiltonian perturbatively 62 causes the stabilizers to drop slightly below 1, indicating the presence of small errors due to higher-order corrections. Nonetheless, the deviation is small as shown in Fig. 5.10. The blue and the green curves are the cases when the system suers from the 1/f Hamiltonian errors dened in Eq. (5.34), where the sum is over all physical qubits (including the monitor qubits). The blue includes continuous indirect detection while the green does not. The sup- pression of errors is apparent, although it is slightly less eective than the ideal 6-qubit case shown in Fig. 5.7a. For most trajectories where errors are suppressed, the stabilizer values stay close to 1. For some trajectories where errors causehS z;x i to ip to1, we can detect them by observing I a;c (t) decaying towards 0 orhS z;x i ^ ipping to1. These behaviors are essentially the same as in Fig. 5.5. 63 Chapter 6 Conclusion 6.1 Conclusion for the lossy measurement model It has been previously shown that any generalized measurement can be performed as a sequence of weak measurements if one has the ability to perform any weak measurement. If only a limited family of weak measurements is possible, however, the class of generalized measurements that can be done may also be limited. This has been explored for dierent plausible families of weak measurements. In Sec. 2, we have signicantly generalized that earlier work by having the weak measurements be destructive: at long times, the system always goes to a xed statej0i. The measurements that can be done in this case must obviously also be destructive: POVMs, rather than ideal generalized measurements. The model of destructive weak measurement captures the behavior of a destructive strong measurement that leaves the system in a xed nal state after the measurement regardless of the outcome. The only freedom in this model is the ability to choose any projective positive- operator-valued measurement on the ancilla in an adaptive way. In Sec. 3, it is shown that, in spite of its limitations, this model can achieve any projective measurement for qubits, and the result can be generalized to any positive-operator-valued measurement (POVM) with commuting POVM elements. However, the method used in Sec. 3 does not work for POVMs with non-commuting elements. In Sec. 4, it is shown that any qubit POVM can actually be achieved by this model. The improvement from POVMs with commuting elements to all POVMs requires a more complicated generalization of the previous method. The approach combines classical pre- and post-processing with a continuous measurement that corresponds to a random walk in a simplex. The pre-processing randomly chooses a linearly independent POVM with elements 64 proportional to projectors from the original POVM. This linearly independent POVM is decomposed into a sequence of destructive weak measurements by mapping the evolution of the positive operator to a random walk in a simplex. The vertex that the walk approaches corresponds to the nal outcome of this succession of weak measurements. Finally, the post-processing chooses which result to output, conditioned on the outcome of the quantum measurement process. The probability of each result agrees with the probability of that result in the original POVM. We have shown that a limited model of destructive weak measurements can achieve an arbitrary POVM for qubits. In higher dimensional systems, what measurements can be achieved by such a restricted class of weak measurements is an open question. The scheme of a system interacting with a stream of qubit probes was shown to be restricted to provide measurements with two distinct singular values [15]. Generalizing the ancilla to higher dimensions gives more degrees of freedom, and that may be a possible route to achieve POVMs for higher dimensional systems. 6.2 Conclusion for continuous indirect measurements We have presented and analyzed a method for the continuous measurement of high-weight operators, and applied this to the problem of continuous quantum error detection by the four-qubit Bacon-Shor code. This method includes engineering an interaction Hamiltonian between the system and the continuously measured ancillary qubits. More nontrivially, the Hamiltonian can be eectively built using physically viable two-local interactions, and the measurements on the monitor qubits consist of well-studied single-qubit continuous measure- ments. In general, this detection scheme can be applied to measuring the stabilizers in any quan- tum code. However, as the weight of the stabilizers in a code increases, the diculty of per- forming this detection scheme is also increased. This is because perturbatively constructing the Hamiltonian for the indirect detection requires applying a strong base Hamiltonian. The 65 strength of this base Hamiltonian grows as the weight of the target term increases because these terms would appear at higher orders in the expansion. This is one of the reasons that we apply it to the four-qubit Bacon-Shor code, where the stabilizers are weight four and their values can be obtained by measuring the two-local gauge operators. In this case, the target Hamiltonian can appear in the second order expansion, which is the minimum. The question of how the construction scheme applies to other quantum codes remains open, but it should certainly apply to the 9-qubit and larger Bacon-Shor codes. Two methods are provided for retrieving the measurement outcomes. The estimator approach is computationally hard but may be benecial to theoretical analysis. By contrast, the signal time average is noisier, but more ecient to perform in real time. It is shown that errors with low rates can be detected and (in the non-Markovian case) suppressed. This is in the regime where the indirect detection is eective. For high-rate or high-strength errors that change the system too rapidly, the detection scheme becomes inapplicable. However, if the type of errors can be learned from the experiments, it may be possible to adjust the setup for better performance. Overall, we have presented a new method for measuring high-weight operators using practical experimental resources. This is a step towards practical quantum error-correction for quantum computing. 66 Reference List [1] Yakir Aharonov, David Albert, and Lev Vaidman. How the result of measurement of a component of the spin of a spin-1/2 particle can turn out to be 100. Phys. Rev. Lett., 60:1351, 1988. [2] Charlene Ahn, Andrew C. Doherty, and Andrew J. Landahl. Continuous quantum error correction via quantum feedback control. Phys. Rev. A, 65:042301, Mar 2002. [3] Charlene Ahn, H. M. Wiseman, and G. J. Milburn. Quantum error correction for continuously detected errors. Phys. Rev. A, 67:052310, May 2003. [4] Charlene Ahn, Howard Wiseman, and Kurt Jacobs. Quantum error correction for con- tinuously detected errors with any number of error channels per qubit. Phys. Rev. A, 70:024302, Aug 2004. [5] Dave Bacon. Operator quantum error-correcting subsystems for self-correcting quantum memories. Phys. Rev. A, 73:012340, Jan 2006. [6] S. B. Bravyi and A. Yu. Kitaev. Quantum codes on a lattice with boundary, 1998. [7] Todd A. Brun. A simple model of quantum trajectories. American Journal of Physics, 70(7):719{737, 2002. [8] Bradley A. Chase, Andrew J. Landahl, and JM Geremia. Ecient feedback controllers for continuous-time quantum error correction. Phys. Rev. A, 77:032304, Mar 2008. [9] Yi-Hsiang Chen and Todd A. Brun. Decomposing qubit positive-operator-valued mea- surements into continuous destructive weak measurements. Phys. Rev. A, 98:062113, Dec 2018. [10] Yi-Hsiang Chen and Todd A. Brun. Qubit positive-operator-valued measurements by destructive weak measurements. Physical Review A, 99(6), Jun 2019. [11] Yi-Hsiang Chen and Todd A. Brun. Continuous quantum error detection and suppres- sion with pairwise local interactions, 2020. [12] M. H. Devoret and R. J. Schoelkopf. Superconducting circuits for quantum information: An outlook. Science, 339(6124):1169{1174, 2013. 67 [13] L Diosi. Quantum stochastic processes as models for state vector reduction. Journal of Physics A: Mathematical and General, 21(13):2885{2898, jul 1988. [14] L. Di osi. Continuous quantum measurement and it^ o formalism. Physics Letters A, 129(8):419 { 423, 1988. [15] Jan Florjanczyk and Todd A. Brun. Continuous decomposition of quantum measure- ments via qubit-probe feedback. Phys. Rev. A, 90:032102, Sep 2014. [16] Jan Florjanczyk and Todd A. Brun. Continuous decomposition of quantum measure- ments via hamiltonian feedback. Phys. Rev. A, 92:062113, Dec 2015. [17] Austin G. Fowler, Matteo Mariantoni, John M. Martinis, and Andrew N. Cleland. Surface codes: Towards practical large-scale quantum computation. Phys. Rev. A, 86:032324, Sep 2012. [18] N. Gisin. Quantum measurements and stochastic processes. Phys. Rev. Lett., 52:1657{ 1660, May 1984. [19] Daniel Gottesman. Stabilizer codes and quantum error correction, 1997. [20] Kung-Chuan Hsu and Todd A. Brun. Method for quantum-jump continuous-time quan- tum error correction. Phys. Rev. A, 93:022321, Feb 2016. [21] Kiyosi It^ o. Stochastic integral. Proc. Imp. Acad., 20(8):519{524, 1944. [22] Kurt Jacobs. Stochastic Processes For Physicists. Cambridge University Press, 2010. [23] Kurt Jacobs. Quantum Measurement Theory and its Applications. Cambridge University Press, 2014. [24] Kurt Jacobs and Daniel A. Steck. A straightforward introduction to continuous quantum measurement. Contemporary Physics, 47(5):279{303, 2006. [25] Stephen P. Jordan and Edward Farhi. Perturbative gadgets at arbitrary orders. Phys. Rev. A, 77:062329, Jun 2008. [26] Yasushi Kondo, Yuichiro Matsuzaki, Kei Matsushima, and Jeerson G Filgueiras. Using the quantum zeno eect for suppression of decoherence. New Journal of Physics, 18(1):013033, jan 2016. [27] Alexander N. Korotkov. Continuous quantum measurement of a double dot. Phys. Rev. B, 60:5737{5742, Aug 1999. [28] Alexander N. Korotkov. Selective quantum evolution of a qubit state due to continuous measurement. Phys. Rev. B, 63:115403, Feb 2001. [29] David Kribs and David Poulin. chapter 6, pages 163{180. Cambridge University Press, 2013. 68 [30] Edoardo Milotti. 1/f noise: a pedagogical review, 2002. [31] K. W. Murch, S. J. Weber, C. Macklin, and I. Siddiqi. Observing single quantum trajectories of a superconducting quantum bit. Nature, 502(7470):211{214, 2013. [32] O. Oreshkov. chapter 8, pages 201{228. Cambridge University Press, 2013. [33] Ognyan Oreshkov and Todd A. Brun. Weak measurements are universal. Phys. Rev. Lett., 95:110409, Sep 2005. [34] Ognyan Oreshkov and Todd A. Brun. Continuous quantum error correction for non- markovian decoherence. Phys. Rev. A, 76:022318, Aug 2007. [35] Juan Pablo Paz and Wojciech Hubert Zurek. Continuous error correction. Proc. R. Soc. Lond. A., 454:355{364, January 1998. [36] Gerardo A. Paz-Silva, A. T. Rezakhani, Jason M. Dominy, and D. A. Lidar. Zeno eect for quantum computation and control. Phys. Rev. Lett., 108:080501, Feb 2012. [37] N. Roch, M. E. Schwartz, F. Motzoi, C. Macklin, R. Vijay, A. W. Eddins, A. N. Korotkov, K. B. Whaley, M. Sarovar, and I. Siddiqi. Observation of measurement- induced entanglement and quantum trajectories of remote superconducting qubits. Phys. Rev. Lett., 112:170501, Apr 2014. [38] D. Tan, S. J. Weber, I. Siddiqi, K. Mlmer, and K. W. Murch. Prediction and retrodic- tion for a continuously monitored superconducting qubit. Phys. Rev. Lett., 114:090403, Mar 2015. [39] Ramon van Handel and Hideo Mabuchi. Optimal error tracking via quantum coding and continuous syndrome measurement, 2005. [40] Martin Varbanov and Todd A. Brun. Decomposing generalized measurements into continuous stochastic processes. Phys. Rev. A, 76:032104, Sep 2007. [41] Martin Varbanov and Todd A. Brun. Weak measurements are universal. Phys. Rev. A, 76:032104, 2007. [42] S. J. Weber, A. Chantasri, J. Dressel, A. N. Jordan, K. W. Murch, and I. Siddiqi. Mapping the optimal route between two quantum states. Nature, 511(7511):570{573, 2014. [43] H M Wiseman. Quantum trajectories and quantum measurement theory. Quantum and Semiclassical Optics: Journal of the European Optical Society Part B, 8(1):205{222, feb 1996. [44] Howard M. Wiseman and Gerard J. Milburn. Quantum Measurement and Control. Cambridge University Press, 2009. [45] S. W uster. Quantum zeno suppression of intramolecular forces. Phys. Rev. Lett., 119:013001, Jul 2017. 69 Appendix Correspondence between4 ~ r and4 n1 The picture relating the position indicator~ x24 n1 and its 3-D image~ r in Bloch sphere is presented in this section. The n can only be 2, 3 or 4 in our cases. Recall from Eqs. (4.23) and (4.24), the denition of~ r is ~ r = 1 ~ x~ q n X i=1 x i q i ~ v i ; (6.1) where~ x;~ q24 n1 , q i > 0 for all i, and the~ v i 's are the Bloch vectors corresponding to a set of linearly independent POVM elements. We dene a bijection T ~ q :4 n1 !4 n1 T ~ q (~ x) 1 ~ x~ q 0 B B B @ q 1 x 1 . . . q n x n 1 C C C A ; (6.2) for any ~ q24 n1 , where q i > 0 for all i = 1; ;n. The inverse map is given by T 1 ~ q (~ x) 1 P n i=1 x i q 1 i 0 B B B @ q 1 1 x 1 . . . q 1 n x n 1 C C C A : (6.3) 70 We can dene a vector~ x ~ q T ~ q (~ x)24 n1 associated with the original~ x. It is equivalent to analyze with~ x and with~ x ~ q because of the bijection T ~ q , i.e., we have a solution for~ x ~ q if and only if we have a solution for ~ x. We choose to work with~ x ~ q since it is linearly related to~ r: ~ r = 0 B B B @ ~ v 1 ~ v n 1 C C C A 1 ~ x~ q 0 B B B @ q 1 x 1 . . . q n x n 1 C C C A =V~ x ~ q ; (6.4) where V 0 B B B @ ~ v 1 ~ v n 1 C C C A 3n (6.5) is a 3 by n matrix. The space~ r lives in is 4 ~ r = ~ rj~ r =V~ x ~ q ; 8~ x ~ q 24 n1 ; (6.6) which is a subspace spanned byf~ v 1 ; ;~ v n g. We claim that rank(V ) =n1 forn = 2; 3; 4. The proof is as follows. Since P n i=1 q i ~ v i = 0 by Eq. (4.36), we have ~ q2 Ker(V ). And by the linear independence of the set of POVM elements, E i (~ x) =q i (I +~ v i ~ ); (6.7) we have that for any vector~ c = (c 1 ; ;c n ) such that n X i=1 c i E i (~ x) = 0; (6.8) then~ c = 0. This implies that there is no nontrivial solution~ c satisfying both ~ c~ q = 0 and n X i=1 c i q i ~ v i = 0: (6.9) 71 We can decompose the n-dimensional vector space into Spanf~ qg Spanf~ qg ? . Suppose a nontrivial vector ~ y2 Spanf~ qg ? is also in Ker(V ), i.e., ~ y~ q = 0 and n X i=1 y i ~ v i =V~ y = 0: (6.10) Then there exists a vector ~ c 0 B B B @ y 1 q 1 . . . yn qn 1 C C C A n X i=1 y i ! 0 B B B @ 1 . . . 1 1 C C C A (6.11) satisfying both conditions in Eq. (6.9), and one can easily check that~ c6= 0. This contradicts the fact that there is no nontrivial solution for Eq. (6.9), and hence no vector in Spanf~ qg ? is also in Ker(V ). We have Spanf~ qg = Ker(V ) and therefore dim(Ker(V )) = 1. By the rank-nullity theorem, we have rank(V ) =n 1. Finally, to show that the map V : 4 n1 ! 4 ~ r is one-to-one, we use the following argument. If ~ x 1;2 24 n1 such that V~ x 1 =V~ x 2 , then ~ x 1 ~ x 2 =a~ q2 Ker(V ) for a number a. By the fact that ~ x 1;2 and ~ q are in4 n1 , a must be zero, and hence ~ x 1 = ~ x 2 . The map V :4 n1 !4 ~ r is onto, by the denition of4 ~ r . Hence4 n1 and4 ~ r are isomorphic. We have dim(4 ~ r ) = dim(4 n1 ) = n 1, and for n = 2; 3; 4, they are a line, a triangle, a tetrahedron. Bayes rule relation The approximation in Eq. (5.12) is pure expansion based on the fact that dt is innitesimal. To the rst order ofdt, theH term does not appear. The overall factorN is irrelevant to the 72 ratio betweenp in our argument and we do not need to expand it. LetdI =fdt+dW=(2 p ). We have p (t +dt) = 1 N n p (t) Tr h (t)e 2( dI dt Zm) 2 dt i +O(dt 2 ) o (6.12) = 1 N p (t) Tr (t)e 2 fZm+ dW 2 p dt 2 dt +O(dt 2 ) (6.13) = 1 N p (t) Tr (t)e 2(fZm) 2 dt2 p (fZm)dW e 2 dW 2 p dt 2 dt +O(dt 2 ) (6.14) = 1 N ( p (t) Tr n (t) I 2 (fZ m ) 2 dt 2 p (fZ m )dW + 2 (fZ m ) 2 dt e 2 dW 2 p dt 2 dt o +O(dt 2 ) ) (6.15) = 1 N p (t) h 1 2 p (fhZ m i )dW i e 2 dW 2 p dt 2 dt +O(dt 2 ) ; (6.16) where dW 2 = dt is used. Replacing Z m byhZ m i and going backwards through the above equalities, we get p (t +dt) = 1 N p (t)e 2 fhZmi + dW 2 p dt 2 dt +O(dt 2 ) = 1 N n p (t)e 2( dI dt hZmi ) 2 dt +O(dt 2 ) o ; (6.17) which shows the Eq. (5.12). 73
Abstract (if available)
Abstract
The first part of this thesis focuses on a study of weak measurements. We consider a qubit model of destructive weak measurements, which is a toy version of an optical cavity, in which the state of an electromagnetic field mode inside the cavity leaks out and is measured destructively while the vacuum state |0⟩ leaks in to the cavity. At long times, the state of the qubit inevitably evolves to be |0⟩, and the only available control is the choice of measurement on the external ancilla system. We show that despite the lossy nature of these weak measurements, any POVM can still be achieved by a sequence of these destructive weak measurements. ❧ The second part is an application of continuous measurements for stabilizer measurements in a quantum code. We provide a method to passively detect the value of a high-weight operator using only two-local interactions and single-qubit continuous measurements. This approach involves joint interactions between the system and continuously-monitored ancillary qubits. The measurement outcomes from the monitor qubits reveal information about the value of the operator. This information can be retrieved by using a numerical estimator or by evaluating the time average of the signals. The interaction Hamiltonian can be effectively built using only two-local operators, based on techniques from perturbation theory. We apply this indirect detection scheme to the four-qubit Bacon-Shor code, where the two stabilizers are indirectly monitored using four ancillary qubits. Due to the fact that the four-qubit Bacon-Shor code is an error-detecting code and that the Quantum Zeno Effect can suppress errors, we also study the error suppression under the indirect measurement process. In this example, we show that various types of non-Markovian errors can be suppressed.
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Chen, Yi-Hsiang
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Destructive decomposition of quantum measurements and continuous error detection and suppression using two-body local interactions
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Doctor of Philosophy
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10/19/2020
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destructive quantum measurements,error detection,error suppression,OAI-PMH Harvest,weak measurements
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