Topological quantum error correction and quantum algorithm simulations

Transcription

1 Topological quantum error correction and quantum algorithm simulations David Wang Submitted in total fulfilment of the requirements of the degree of Doctor of Philosophy August 2011 School of Physics The University of Melbourne

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3 Abstract Quantum computers are machines that manipulate quantum information stored in the form of qubits, the quantum analogue to the classical bit. Unlike the bit, quantum mechanics allows a qubit to be in a linear superposition of both its basis states. Given the same number of bits and qubits, the latter stores exponentially more information. Quantum algorithms exploit these superposition states, allowing quantum computers to solve problems such as prime number factorisation and searches faster than classical computers. Realising a large-scale quantum computer is difficult because quantum information is highly susceptible to noise. Error correction may be employed to suppress the noise, so that the results of large quantum algorithms are valid. The overhead incurred from introducing error correction is neutralised if all elementary quantum operations are constructed with an error rate below some threshold error rate. Below threshold, arbitrary length quantum computation is possible. We investigate two topological quantum error correcting codes, the planar code and the 2D colour code. We find the threshold for the 2D colour code to be 0.1%, and improve the planar code threshold from 0.75% to 1.1%. Existing protocols for the transmission of quantum states are hindered by maximum communication distances and low communication rates. We adapt the planar code for use in quantum communication, and show that this allows the fault-tolerant transmission of quantum information over arbitrary distances at a rate limited only by local quantum gate speed. Error correction is an expensive investment and thus one seeks to employ as little as possible without compromising the integrity of the results. It is therefore important to study the robustness of algorithms to noise. We show that using the matrix product state representation allows one to simulate far larger instances of the quantum factoring algorithm than under the traditional amplitude formalism representation. We simulate systems with as many as 42 qubits on a single processor with 32GB RAM, comparable to amplitude formalism simulations performed on far larger computers.

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5 Declaration This is to certify that i. the thesis comprises only my original work towards the PhD except where indicated in the Preface, ii. due acknowledgement has been made in the text to all other material used, iii. the thesis is less than 100,000 words in length, exclusive of tables, maps, bibliographies and appendices.

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7 Preface Chapter 1 is an original review, and Chapters 2 5 are based on published material[1 5]. Technical details and software used in simulations, unless otherwise stated, were developed by myself. This work benefited greatly from insightful advice from the following contributors: Charles Hill (Chapters 5 and 6), Ashley Stephens (Chapter 2), and Zachary Evans (Chapter 2).

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9 Acknowledgements To my supervisors, Lloyd Hollenberg and Austin Fowler, for their diligent reading and their continual support. My colleagues Ashley Stephens, Zachary Evans, and Charles Hill, who have been a wealth of knowledge and provided endless discussions. To my friends Daniel, James, Michael and Nadine. And of course, to my Family. All of you made this possible.

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11 Contents 1 Introduction Quantum Computing and Quantum Algorithms Quantum Mechanics of Qubits The Stabiliser Formalism Quantum Error Correction Circuit Notation Topological Quantum Computation Non-Abelian Anyons The ν = 5 Quantum Hall State Computation with Anyons Qubit Architectures Ion Traps Superconducting Qubits Quantum Dots The Planar Code The Toric and Planar Codes Syndrome Extraction Error Correction Graph Matching Example Time to Failure Simulations and Results Conclusion Enhanced Planar Code Error Recovery Accurate Error Propagation Non-integral Link Weights Results Conclusion i

12 Contents 4 Surface Code Quantum Communication Motivation Quantum Communication Protocol Communication Link Errors Bell Pair Error Rate Loss of Bell Pairs Combined Bell Pair Errors Conclusion The 2D Colour Code Motivation Error Correction on the 2D Colour Code Logical Qubits and Logical Gates Hypergraph Mimicry and Pair Assignment Ideal Threshold Error Rate Direct simulation Dangerous syndrome coverage Simulation Results Conclusion Order-Finding and Matrix Product States Review of Shor s Algorithm Matrix Product States and Space Complexity Implementation and Benchmarks Multi-Process MPS Conclusion Summary and Conclusions 103 ii

13 List of Figures 1.1 Circuit representations for applying (a) a single-qubit unitary Û; (b) a cnot gate; and (c) a swap gate Circuit representations for measurement (a) and (b) in the Z basis; (c) in the X basis A non-fault-tolerant circuit preparing the 4-qubit cat-state Arrangement of qubits on a torus with nearest-neighbour interactions. The stabiliser generators are the tensor products of Z and X on the qubits around faces and intersections respectively Examples of error syndromes on the toric and planar codes The two logical-x and logical-z operators in the toric code Eliminating the toric code s periodic boundaries to produce the 2D planar code Circuits used to measure (a) X stabilisers, and (b) Z stabilisers. Ancilla qubits have at most four neighbouring data qubits, which are denoted by their respective compass directions Sequence of cnot gates permitting simultaneous measurement of all stabilisers D syndrome formed when using non-ideal syndrome extraction circuits The syndrome observed depends only the terminals of error chains, thus is not unique The toric code error correction methodology adapted for the planar code and its boundary conditions Optimisations arise from the inclusion of boundary nodes Example syndrome on the toric code, and the associated graph for error correction Example of a matching Example of an alternating path iii

14 List of Figures 2.14 Example of a perfect matching Average time before a state sustains a logical failure, assuming error-free syndrome extraction. p th = ± Average time before a state sustains a logical failure, using the syndrome extraction circuits of Fig p th = Two errors artfully arranged to cause premature failure under the standard Manhattan distance metric Example of propagation of data errors when one syndrome is measured more frequently than another Average time before the logical qubit is corrupted when (a) the standard measure of separation of syndrome changes is used, and (b) accounting for error propagation by two-qubit gates Circuits determining the sign of (a) XXXX stabiliser, and (b) ZZZZ stabiliser without explicit initialisation gates Observed syndrome changes as a result of a single error Numbered error processes from Fig. 3.5 contributing to specific links Average time before logical failure when using d max (s 1,s 2 ) and the standard error model. p th = 1.1% Average time before logical-x failure when using d 0 (s 1,s 2 ) and the standard error model. p th = 1.1% (a) Entanglement swapping. (b) Entanglement purification Surface code quantum communication protocol The surface code lattice is divided into columns and distributed amongst a linear chain of quantum repeaters that are connected by Bell pairs Measurement of stabilisers spanning two repeater stations Average number of error correction rounds before logical failure versus Bell pair error rate p B and code distance d Average number of error correction rounds before logical failure versus loss p loss and code distance d Probability of logical error per link for a variety of loss and Bell error rates iv

15 List of Figures 5.1 State distillation circuits for the Y and A magic states, used in performing S and T respectively D lattice of qubits for the colour code. The stabiliser generators are the tensor products of X and Z on the qubits around each plaquette Examples of syndromes observed on the 2D colour code Examples of syndromes observed on finite lattices The hypergraph constructed from the observed syndrome. The hypergraph matching identifies a corresponding set of error chains which produces the observed syndrome Triangular lattices permitting all Clifford group gates to be performed transversely Examples of red and blue defects on the colour code If all of the hyperedges present in the minimum-weight hypergraph matching are known, one can solve the matching problem efficiently If two of the nodes from each of the hyperedges present in the minimum-weight hypergraph matching are known, one can solve the matching problem efficiently Construction of the mimic graph, a graph which incorporates some of the properties of the original hypergraph. The nodes ḡ, p, p, and b are introduced for the future inclusion of hyperedges Incorporating red nodes and hyperedges from the original hypergraph into the mimic graph Partial patterns observed in the matching of mimic graphs and their interpretations as corrections Malformed patterns in a mimic graph matching can often be removed by adding a weight-0 alternating cycle Average number of error correction rounds before logical failure under error-free syndrome extraction. p th = 13.3% The leading contribution to the logical error rate, A d (F), when using minimum-weight hypergraph matching grows exponentially with the distance d of the code The number of dangerous syndromes as a result of exactly F +k errors when using the approximate matching method grows as O ( 6 (d 1)/2 2 ( )) d Q F k v

16 List of Figures 5.17 Rules for shifting terminals amongst same colour plaquettes Analytic time to logical failure under error-free syndrome extraction when (a) first order terms only, and (b) higher order terms are used to calculate the logical error rate. p th = 6.25% Circuit to fault-tolerantly prepare a four qubit cat-state, which is used to measure the parity from octagonal plaquettes The average time to failure and the logical error rate of a quantum memory. p th = 0.10 ± 0.01% Schematic of the order-finding circuit which efficiently determines the order parameter r The probability distribution for the states in the upper register after completing the order-finding circuit. High probability densities are observed around integer multiples of 2 2l /r One possible MPS representation of ψ = 1 2 ( ) Quantum Fourier transform under MPS vi

17 List of Tables 6.1 Internal qubit order is important in MPS as only neighbouring qudits may interact with one another. Qubits are reordered as shown to apply the indicated gate. Qubit and gate lettering correspond to those in Fig The ranks in the MPS state after applying all gates up to and including the indicated controlled-unitary for N = 65, x = 2, l = 7. The second last number in each row governs the size of the lower register, and the execution time of the unitary. The underlined value is the order parameter r. The final order of qubits is q 0,q 1,,q Applying the controlled-unitaries in decreasing powers of two lead to equal or lower ranks across the MPS than before, thus improves performance. The case in Table 6.2 can be completed using only a fraction of the memory and time. The final order of qubits is q 13,q 12,,q CPU time in seconds to execute the order-finding circuit (Fig. 6.1) from the start to: t 1 complete all controlled-us; t 2 measure out the lower register; and t 3 complete the entire circuit. x is chosen to maximise the order r for the given N. r determines the size of the matrices before the QFT vii

18 List of Tables viii

19 1 Introduction 1.1 Quantum Computing and Quantum Algorithms Quantum mechanical systems are inherently difficult to simulate on a classical computer because the Hilbert space grows exponentially for every additional particle. The classical computer, whose bits can take values 0 or 1, requires exponentially increasing space to represent the state. Feynman proposed that a quantum computer a computer that manipulates qubits[6], quantum bits, two level quantum systems that also reside in Hilbert space and can be in linear superpositions of the states 0 and 1 may be able to simulate such systems more effectively[7, 8]. The discovery of quantum algorithms[9 12] capable of outperforming existing classical algorithms has accelerated interest in quantum computation. Shor s algorithm for integer factorisation[9, 10], in particular, has drawn much attention. It shows that a quantum computer can factorise numbers efficiently, that is, using only polynomially growing resources, and is exponentially faster than all known classical algorithms. The Rivest-Shamir-Adleman[13] protocol for public-key cryptography is founded on the assumed difficulty of factorising large numbers classically. A large-scale quantum computer therefore compromises security systems based on these techniques. Algorithms stemming from the complexity of the discrete logarithm problem are similarly compromised. There are also quantum algorithms to perform general searches[11, 12], to simulate quantum systems and determine chemically interesting quantities[14 16]. Unfortunately, large-scale quantum computers will inevitably suffer from some form of decoherence, adding to the challenge of realising a quantum computer capable of outperforming a classical computer. One can counteract this by investing in error correction applicable to quantum computing. The existence of quantum error correcting codes[17 21] is curious in light of three 1

20 1 Introduction formidable difficulties: the no-cloning theorem[22, 23] dictates that is impossible to copy quantum states; the need to correct for continuous errors, not just discrete errors as in classical error correction, as qubits are fundamentally analogue devices; and finally, a measurement will in general collapse the wavefunction thereby altering the state. Nevertheless quantum error correcting codes exist, which improves the prospects for a large-scale quantum computer. Other forms of noise reduction such as dynamic decoupling exist, however only quantum error correction is able to handle general noise. The quantum accuracy threshold theorem[24 29] states that when quantum error correction is used in conjunction with circuits that spread errors in a limited manner (i.e. fault-tolerant circuits[30 32]), one can efficiently perform arbitrarily large quantum computations provided that all gates fail with probability below some threshold error rate p th. (The use of fault-tolerant circuits incurs only polylogarithmic overhead.) To see this, imagine that encoding physical qubits into logical qubits suppresses the logical error rate (i.e. below threshold). One can repeat this process, encoding lower-level logical qubits into successively higher-level logical qubits. This is known as concatenation, and the similarity of the process ensures that each iteration reduces the logical error rate. Early calculations estimated a threshold value of approximately p th = 10 6 [25 28]. While physically realising quantum operations with such a stringent error rate is highly impractical, all these ingredients combined show that large-scale quantum computation is feasible in principle, even when the system suffers from decoherence. The [[7, 1, 3]] Steane code[20] and the [[9, 1, 3]] Bacon-Shor code[33] are well known examples of concatenated codes. More detailed calculations find that these schemes possess threshold error rates p th = [34] and p th = [35] respectively. In many physical implementations, it is unreasonable to interact arbitrary pairs of qubits without regard of the physical separation, and yet maintain a fixed error rate. The threshold values for these schemes are diminished to p th = [34] and p th = [36] when the qubits are arranged on a regular 2D lattice supporting only nearestneighbour interactions. In this Thesis, we shall investigate a different family of error correcting codes, namely topological codes[37]. By far the most prominent of the topological codes is the planar code, often called the (2D) surface code[37 46]. This code can be implemented on a regular 2D lattice of qubits with nearest-neighbour 2

21 1.1 Quantum Computing and Quantum Algorithms interactions, and has a relatively high threshold error rate of p th 0.75%. We shall introduce this code formally in Chapter 2, describing in detail its error correction methodology which is used to provide an independent verification of the threshold. Chapter 3 further refines this work, giving rise to quadratic improvement in performance for low error rates, and a threshold error rate p th = 1.1%. Other schemes with threshold values in excess of 1% have been devised in recent times[47 49], however this is the highest threshold calculated in a geometrically constrained setting. In Chapter 4, the 2D planar code is adapted for use in transmitting quantum states over large distances. We describe a protocol that allows the faulttolerant transmission of arbitrary quantum states over a linear chain of repeater stations connected by Bell pairs. The protocol requires only one-way classical communication, allowing quantum information to be transmitted at a rate limited by the local gate speed. In Chapter 5 we look at a different topological code, the (4.8.8) colour code[50, 51]. The code is interesting as it can perform one of the operations required for universal quantum computation more simply than in the 2D surface code; in general, the logical operation Ū for the logical qubit is not simply obtained by applying the physical operation Û to every physical qubit. The colour code features a more complex underlying lattice and circuits, and as a result, we find its threshold is lower, p th 0.1%. Demanding excessively large distance codes to perform reliable computation is impractical. Instead, one strives to use as little error correction as possible to achieve the same goal. It is therefore important to study the tolerance of algorithms to errors. However, as already mentioned, simulating large quantum algorithms on classical computers is difficult because the memory requirements grow too rapidly, making such assessments difficult. In Chapter 6, we show that the symmetries in the factoring problem reduce the space complexity when the state of the system is represented as a matrix product state (MPS)[52]. This reduction allows systems with as many as 36 qubits to be simulated on a laptop with 2GB of memory. Simulations involving up to 42 qubits are possible using a single processor of a server with 32GB of memory. The remainder of this Chapter will be dedicated to review material. Section 1.2 gives an overview of quantum mechanics. Section 1.3 reviews the stabiliser formalism. We introduce a simple bit-flip code using this formalism in Section 1.4. We review the circuit notation and some common gates in Section

22 1 Introduction Section 1.6 introduces the topological model for quantum computing, from which topological quantum error correction is inspired. Finally we examine some common architectures in Section 1.7. The final two sections provide background knowledge for the interested reader only, and are not necessary for the work in the Thesis. 1.2 Quantum Mechanics of Qubits The qubit is the fundamental unit of information with which quantum computers perform computation. Quantum mechanics allows a single particle with wavefunction ψ to be in a linear superposition its basis states. Thus a general single qubit state can be written [ ] α ψ = α 0 + β 1. (1.1) β Here α and β are complex coefficients satisfying α 2 + β 2 = 1. α 2 and β 2 are equal to the probabilities of measuring the particle in the 0 and 1 states respectively. A system of n qubits has basis spanned by { 0, 1 } n, where is the Kronecker tensor product. Thus an n qubit register has 2 n basis states and stores 2 n complex coefficients: ψ = 2 n 1 i=0 α i i. Here the is are the integer representations of the product states. Evolving the system according to some Hamiltonian H for time t generates a unitary operation Û. For the special case of a time-independent Hamiltonian H, the Schrödinger equation gives Û = e iht. (1.2) The unitary operation Û acts on all 2n basis states simultaneously to give ψ = Û ψ = 2 n 1 i=0 α i Û i = 2 n 1 i=0 α i i. Because a function (Û) can be simultaneously calculated for many input parameters (the initial superposition s basis states), this is usually known as quantum parallelism. Unfortunately, this looks more promising than it actually is. The reason is this: in order to extract information from the system, one performs a measurement which collapses the wavefunction. For example, let us measure the least significant qubit in the register ψ, which we rewrite as ψ = 2 n 1 i even ( ) α i i/2 0 + α i+1 i/2 1. (1.3) 4

23 1.3 The Stabiliser Formalism That qubit will probabilistically result in 0 or 1, leaving 2n 1 ψ measure 0 ψ 0 = A 0 i even i even α i i/2 0, ψ 0 ψ 0 = 1 (1.4) 2n 1 ψ measure 1 ψ 1 = A 1 α i+1 i/2 1, ψ 1 ψ 1 = 1. (1.5) A 0 and A 1 are normalisation constants to retain the probability interpretation of the coefficients. Thus measuring the entire register will yield just one of the 2 n possible states. Note that measurement does not give the complex coefficient of the measured state. It is tricky to devise algorithms to properly exploit this parallelism. Nevertheless, quantum algorithms capable of outperforming existing classical algorithms have been discovered[9 12]. 1.3 The Stabiliser Formalism Explicitly writing out the entire quantum state of a system in Dirac notation soon becomes tedious for keeping track of states important in error correction. It is far more convenient and descriptive to work using stabiliser notation[53]. A stabiliser S of a state ψ is defined to be an operator S ψ ψ. Since the product of many stabilisers is also a stabiliser, it is redundant to state all stabilisers. Instead, a general n qubit state can be described by a set of n independent commuting stabiliser generators. Calderbank-Shor-Steane (CSS) error correction codes[17 20] have logical states 0 and 1 defined by stabilisers consisting of tensor products of identity I and the Pauli operators σ X, σ Y, and σ Z only. Thus they are defined much more succinctly in terms of their stabiliser generators. The abundance of the Pauli operators in error correction makes it convenient to define X, Y, and Z: [ ] [ ] [ ] i 1 0 X σ X =, Y σ Y =, Z σ Z =. (1.6) 1 0 i X is the bit-flip operation ( 0 1 ), Z is the phase-flip ( 1 1 ), and Y = ixz. The Pauli matrices have well known algebraic properties. σ i = σ i, i (1.7) σ 2 i = I, i (1.8) σ i σ j = σ j σ i, i j (1.9) 5

24 1 Introduction Consider a state ψ described by its generators {S i }. Then, the state ψ after applying the unitary operation Û on ψ is stabilised by S i = ÛS iû, i: ) ) S i ψ = (ÛSi Û (Û ψ ( ) = Û S i ψ = Û ψ = ψ. (1.10) As an example, let us apply Û = Z to the state stabilised by {+X}, + = 1 2 ( ). (1.11) The original stabiliser generator is transformed as described above, and simplified using the Pauli algebra: Z(+X)Z = ZZ X = X. Thus the resultant state is stabilised by { X}. This is the state, = 1 2 ( 0 1 ). (1.12) 1.4 Quantum Error Correction Consider the three qubit bit-flip code[6] which has logical states 0 = 000 and 1 = 111. Let the qubits in the state be denoted by q 1, q 2, and q 3. The stabiliser generators for the code are S 1 = +Z q1 Z q2 I q3, (1.13) S 2 = +I q1 Z q2 Z q3. (1.14) We will suppress the subscripts and simply use the juxtaposition to denote the qubit each term acts upon hereafter. Here we have specified one fewer generator than qubits, thus the state is not fully defined. This is typical in error correction as it introduces a degree of freedom to store the logical state. For example, S 3 = ±ZZZ would encode 0 and 1. Since generators are simply operators, their expectation values can be measured. The generator encoding the logical state (i.e. S 3 ) must not be measured during error correction; that would collapse the encoded state. The remaining generators, however, simply provide the redundancy to perform error correction. For example, measuring S 1 = ZZI gives ψ S 1 ψ = ψ (+ZZI) ψ = +1. (1.15) 6

25 1.4 Quantum Error Correction Consider now a bit-flip error on the first qubit, producing ψ = (XII) ψ. Measuring S 1 again now gives ψ S 1 ψ = ψ (XII)(+ZZI)(XII) ψ (1.16) = ψ ( ZZI) ψ (1.17) = 1. (1.18) That is, we can measure the eigenvalues of these generators. These values change sign depending on the errors incurred, thus form the error syndrome. By enforcing the state to be in the +1 eigenstate of these generators, we return the state to a linear superposition of the code words. Notice that the logical operation X = XXX commutes with all of the generators in Equation 1.14, but not S 3 ; S 3 will instead change sign to reflect the change in logical state as desired. Likewise, a bit-flip error on each of the three qubits in the code will perform a logical operation. The minimum number of elementary operations required to undetectably alter the logical state is the (Hamming) distance d of the code. An error correcting code can reliably correct (d 1)/2 or fewer errors. The three qubit bit-flip code can correct single X errors. However, this code provides no mechanism for detecting Z errors; Z errors commute with every operator measured. Thus it cannot protect general quantum states. In order to simulate quantum codes suffering logical failure, we shall assume the system suffers depolarising noise. That is, all fundamental operations are assumed to take unit time and have some probability p of failing. Initialisation will prepare the desired state, 0 or +, with probability (1 p), and the erroneous state, 1 or, with probability p. Similarly, a measurement have a probability p of reporting the wrong result. A single-qubit gate Û is applied perfectly, followed by an equal probability p/3 of that qubit incurring each of the X, Y, and Z errors. Idle qubits are also subject to this type of noise model. We will abbreviate this statement to the equation (1 p)i + p (X + Y + Z). (1.19) 3 A two-qubit gate is also applied perfectly, now followed by one of the 15 non-trivial tensor products of I, X, Y, and Z to account for correlated errors. (1 p)ii + p (IX + IY + IZ + XI + XX + XY + XZ 15 + Y I + Y X + Y Y + Y Z + ZI + ZX + ZY + ZZ). (1.20) 7

26 1 Introduction Note that the above equations are not quantum operations; they are merely statements describing the probabilities of each error combination arising. For a state with density matrix ρ, the single-qubit case is expressed as E(ρ) = (1 p)ρ + p (XρX + Y ρy + ZρZ). (1.21) Circuit Notation We shall work in the circuit model for quantum computation[54, 55]. Other models for computation exist such as quantum Turing machines[56], topological quantum computation[37], adiabatic quantum computation[57], and cluster states[58, 59]. A quantum circuit diagram is read like a piece of music, with time flowing from left to right. At the left, one may specify the initial state of the system, and at the right, the final state using ket notation. The horizontal reference lines represent the path of qubits. Many qubits may be grouped together to form a single register. These are represented by the slash through the reference line. They can also represent a qudit, the d-dimensional counterpart to the 2- dimensional qubit; there is no mathematical difference between the two cases. Symbols appearing on reference lines denote operations acting on those qubits. A single qubit gate Û is shown as Û inside a rectangle (Fig. 1.1a). Similarly, multi-qubit gates are shown as a rectangle encompassing all involved qubits. There are a few special cases. A controlled-unitary takes a control qubit and applies the unitary Û to the target register only when the control qubit is in the 1 state. This is represented by a filled circle on the control qubit, connected to the unitary gate by a vertical line. Furthermore, when the unitary is X, this is known as the controlled-not (cnot) gate, and the target unitary is instead drawn with the especial xor symbol ( ). A cnot gate is shown in Fig. 1.1b. Another common gate is the two-qubit swap gate, which exchanges the values of two qubits: a b b a. It is shown as two crosses connected by a vertical line (Fig. 1.1c). Finally, a measurement is performed in the computational basis (i.e. Z basis) unless otherwise specified. It is represented by a meter, the gate M X or (Fig. 1.2). Additionally, qubits which have no further operations to perform can be implicitly assumed to be measured. Popular gates encountered are those in the standard set of gates for universal quantum computation: X, Y, Z, Hadamard (H), phase (S), π/8 (T), 8

27 1.5 Circuit Notation a Û Û a a, b a, a b a, b b, a (a) (b) (c) Figure 1.1: Circuit representations for applying (a) a single-qubit unitary Û; (b) a cnot gate; and (c) a swap gate. a 0/1 (a) a 0/1 (b) a M X (c) +/ Figure 1.2: Circuit representations for measurement (a) and (b) in the Z basis; (c) in the X basis. Possible outcomes are shown to the right. and controlled-not (cnot). One also commonly finds the controlled-sign gate (csign or CZ) in place of the controlled-not gate. This is a finite set of gates which together give the ability to perform arbitrary rotations, analogous to the universality of the nand gate in classical logic. Other universal sets of gates exist[60 62]. Assuming the computational basis states arranged in increasing order 0, 1 for single-qubit gates, and 00, 01, 10, 11 for two-qubit gates with the control qubit as the most significant bit, the standard set of gates are described by the following matrices: [ ] [ ] [ ] H = , S =, T =, (1.22) i 0 e iπ/4 [ ] [ ] [ ] [ ] I cnot = = I , csign = = (1.23) 0 X Z The T gate is also known as the π/8 gate because it is equivalent to having e ±iπ/8 along the diagonal, up to an unimportant global phase. The inclusion of the π/8 gate in the set is most important; the other gates form the Clifford group, and any circuit consisting of only these operations can be simulated efficiently using the stabiliser formalism. The π/8 gate is an element outside of the group, thus gives rise to the ability to perform arbitrary rotations. Equally, the single-qubit rotations X, Y, Z, and S are often more easily implemented in practice as elementary operations than as products of H and T, thus are retained in the standard universal set of gates. As an example of a quantum circuit, see Fig. 1.3 which prepares a 4-qubit cat-state, an even superposition of the all 0 and all 1 states. The name invariably prompts explanation, and this certainly is no exception. It is an 9

28 1 Introduction 0 H ( ) Figure 1.3: A non-fault-tolerant circuit preparing the 4-qubit cat-state. allusion to Schrödinger s cat either being dead after inhaling hydrocyanic acid, or still alive and kicking. ψ cat = 1 2 ( ) (1.24) Note that this is very different from applying H on every qubit on the all 0 state, which instead produces an equal superposition of all states. ψ H n = 1 2 n 1 i (1.25) 2 n To see that the circuit in Fig. 1.3 is not fault-tolerant, we insert an X error on the third qubit at the start of the circuit. It is easy to prove that a cnot gate will propagate X errors from the control to target, and Z errors from target to control. Thus the X error propagates to the fourth qubit, resulting in two errors in the cat-state. The assumption here is that the cat-state will then interact with a single encoded logical qubit, thus resulting in these two i=0 errors being propagated to the same encoded block. 1.6 Topological Quantum Computation We give a short review the topological model for quantum computation[37]. Studying different models of computation is can be insightful, leading to new algorithms or, as in our case, new methods of error correction. It has been shown that the topological model can have equivalent power to the circuit model[63, 64]. That is, one can efficiently translate between the two formalisms. A review of topological quantum computation can be found in[65]. Let us clarify that a topological quantum computer is a piece of hardware with its own primitive operations. The way these operations are performed provides a topological quantum computer with its own form of protection from errors (i.e. protection at the hardware level). This is very different from topologically based quantum error correction, which is a general error 10

29 1.6 Topological Quantum Computation correction scheme that draws inspiration from topological quantum computers, but may be applied to other quantum computer architectures. Our work in quantum error correction focuses on the latter, thus the language and content introduced here is exclusive to this Section, and the reader need not be familiar with the content for the work in the Thesis Non-Abelian Anyons Topological quantum computers are built from particles known as anyons. We first motivate the existence of particles in 2 + 1D systems with obeying different statistics to bosons and fermions, following[66]. We shall discuss physical realisations of anyons later. Consider adiabatically interchanging two indistinguishable particles twice. This is equivalent to looping one particle around the other. In a 3+1D system (three spatial and one time dimension), the process returns the system to the initial state, thus a single interchange of particles can change the wavefunction ψ ± ψ. This corresponds to the particles being bosons or fermions, which obey bosonic or fermionic statistics respectively. A 2 + 1D system is inherently different from a 3 + 1D system as the loop formed cannot be contracted to a point without crossing through the other particle; performing a loop is topologically different from having not performed a loop. One can distinguish between winding the particle clockwise or counterclockwise around the other, so the entire process need not return the system to the initial state. Instead, a single particle interchange can pick up an arbitrary phase to the wavefunction ψ e iθ ψ, for θ R. Particles obeying this type of statistics, known as anyonic statistics, are known as anyons. The process of interchanging two anyons is known as braiding. Computation is performed in the degenerate ground state of a system of anyons. Anyons are arranged in the line, and braiding neighbouring pairs of anyons is the only way to perform unitary operations on a topological quantum computer. Anyons are called non-abelian if two braiding operations do not commute. The use of abelian anyons allow implementations of quantum memories, from which topological error correction is derived, whereas non-abelian anyons are capable of universal quantum computation. Topological quantum computers protect against errors due to primitive operations corresponding to braids; an operation is performed when a braid is 11

30 1 Introduction formed, which is a Boolean quantity. Therefore topological quantum computers do not suffer from imprecision due to the intrinsically analogue nature of qubits. Quantum information is said to be encoded non-locally in topological quantum computing braids are not quantities associated with individual particles making it robust against systematic and random rotation errors (e.g. rotating a single-qubit about the X-axis by π), and local interactions with the environment. Instead, an operation is only unwittingly performed if thermal excitations spontaneously create pairs of quasiparticles which drift apart, perform a braid, before finally annihilating. Provided that the topological quantum computer is run at a sufficiently low temperature, the energy gap between the ground state and the excited states will exponentially minimise the chance of this occurring The ν = 5 2 Quantum Hall State At low temperatures, when a current runs perpendicular to a large external magnetic field, the diagonal resistivity ρ xx = 0 and the Hall resistance ρ xy = 1 h is quantised with ν Z. Here h is Planck s constant and e is the electron ν e 2 charge. This phenomenon is known as the integer quantum Hall effect. Later experiments performed by Tsui et al. discovered the existence of these same characteristics at a filling factor of ν = 1 [67]. This is known as the fractional 3 quantum Hall effect. In particular, early experiments observing the fractional nature of ν = p found odd-denominators q. These are explained by Laughlin s q many-particle wavefunction[68] which follows from the antisymmetry of the wavefunction under particle exchange. However, Willett et al. observed the existence of even-denominator (ν = 5 2 ) fractional quantum Hall states[69] which Laughlin s theory does not account for. Moore and Read were the first to explain this effect[70]. In their theory, the ν = 5 quantum Hall states can be described by quasiparticles (anyons) with 2 fractional charge e = e/4 which obey non-abelian statistics. Experiments have observed quasiparticles with charge e = e/4[71 73]. Experiments have also been proposed to determine whether these quasiparticles obey Abelian or non-abelian statistics[74 76]. However, these experiments have yet to be conducted. If these quasiparticles do indeed obey non-abelian statistics, then they would be a suitable candidate for constructing a topological quantum computer. 12

31 1.7 Qubit Architectures Computation with Anyons The excitations of the ν = 5 state are known as Ising anyons. There are three 2 species of anyons: 1, σ, and ψ. Two particles brought into proximity will fuse to produce a new particle. The fusion process obeys an arithmetic similar to the addition of spin- 1 particles. In particular, two σ anyons 2 ( charge-1 ) can 2 fuse to give a 1 anyon ( charge-0 ), or a ψ particle ( charge-1 ). The model is non-abelian as two particles can fuse in more than one way. The species of the resultant particle differ in their topological charge which can be measured. A simple qubit can be created from a pair σ anyons: the qubit is in 0 if the two σ anyons fuse to give 1, and 1 if they fuse to give ψ. However this representation has some drawbacks which we shall not discuss. A qubit can be created from four σ anyons as discussed in[66]. However, one finds that by braiding Ising anyons, one can only perform one-qubit π/2 rotations about the X and Z axes. Since circuits using these operations only can be simulated efficiently classically using the stabiliser formalism, universality is not achieved though braiding alone; one must supplement these operations with, for example, a single-qubit π/4 rotation (e.g. T) and a two-qubit entangling gate (e.g. cnot or csign). These can be realised through the use of non-topologically protected operations and magic state distillation[66, 77, 78]. Another possible method to achieve universality is through dynamical topology changing[79]. Computation using other types of anyons follows similarly. In particular, Fibonacci anyons[64, 65] are capable of universal quantum computation exclusively through braiding[80]. While there are no known natural systems that realise such excitations, one can artificially construct string-net models with such anyons. However, simple operations correspond to complex braids. 1.7 Qubit Architectures There are many approaches to realising qubits. We have already mentioned the possibility of creating qubits from pairs of Ising anyons. Just as anyons and hence topological quantum computers are realised in 2D systems, we shall see that the simplest topological quantum error correcting code, the planar code, is naturally implemented on 2D architectures. Some common architectures for realising more conventional qubits include ion traps[81], superconducting 13

32 1 Introduction qubits[82 90], quantum dots[91], optical lattices[92, 93], photonic modules[94], N-V centres[95, 96], and donors in silicon[97, 98]. We describe some physical realisations below for concreteness only. In the subsequent Chapters, the exact architecture is irrelevant; it is relatively easy to specialise the error model to suit a particular case. Thus, in this work, qubits will naturally emerge as purely mathematical objects obeying the laws of quantum physics Ion Traps For ion traps[81], a qubit can be encoded in the energy level of an ion; for example, the ground state corresponds to 0 and excited states correspond to 1. A laser directed at an ion allows that qubit to transition between its basis states. The Coulomb repulsion between ions allow interaction within qubits via the exchange of a phonon. Experiments have achieved single-qubit gates with error rate 2.0(2) 10 5 [99], two-qubit unitaries with fidelity 99.3(1)%[100], and readout with error rate 0.9(3) 10 4 [101]. Furthermore, states with up to 14 entangled qubits have been created[102] Superconducting Qubits There are three main types of superconducting qubits: charge[82 86], flux[87, 88] and phase[89, 90] qubits. All three cases make use of Josephson junctions, thin insulating barriers between two superconducting metals. Josephson junctions give rise to systems with nonlinear separation between energy levels, necessary so that one does not, for example, drive transitions between the first and second levels when only transitions between the ground and the first levels are intended. The different types are distinguished in the variable used to control quantum state. Reviews of superconducting qubits can be found in[ ]. Single-qubit operations and readout have been demonstrated in charge qubits[106, 107], flux qubits[108, 109], and phase qubits[110, 111]. Two-qubit operations have been demonstrated in charge qubits[112, 113], flux qubits[114, 115], and phase qubits[116, 117]. 14

33 1.7 Qubit Architectures Quantum Dots A particle s spin is one of nature s two level states. As an example, a qubit can be encoded in the spin of the excess electron of a single-electron quantum dot[91]. Single-qubit rotations can be realised through electron spin resonance[118] or optical pulses[119]. Tunable potential barriers between neighbouring quantum dots allow one to control the tunnelling between quantum dots and hence control two-qubit interactions[120]. One can also create qubits from the electron charge state in double quantum dot systems[121, 122]. 15

34 1 Introduction 16

35 2 The Planar Code In this Chapter we shall introduce the toric and planar codes[37], which differ only by boundary conditions, following the ideas of[38]. The planar code is often referred to as the (2D) surface code, and is a representative of topological quantum error correcting codes, the main subject of this Thesis. The concepts introduced in this Chapter thus form the foundation for the work to come. For any error correcting code, the recovery phase requires one identify a likely set of errors with a syndrome consistent with observation. In other words, one must reproduce the observed syndrome. We achieve this via the minimumweight perfect matching algorithm from graph theory[ ]. Renormalisation techniques exist capable of processing perfect syndrome information[127]; however, at present only the minimum-weight perfect matching algorithm can be used to process the output of realistic quantum circuits. Whilst we shall adapt and modify the input graph for different codes and to improve the code s performance, the underlying methodology will remain the same as introduced here. Section 2.1 is an original review of the toric and planar codes. Section 2.2 presents the syndrome extraction circuits used throughout the simulations. An efficient method to correct errors is described in Section 2.3, and an extended example is provided in Section 2.4. Section 2.5 describes the simulation details, which yields an independent verification of the accuracy threshold to be p th = 0.75%[43]. The verification is important as the original threshold was found to be considerably higher than that of concatenated codes on similar 2D nearestneighbour architectures; the 7-qubit Steane code has p th = [34], while the 9-qubit Bacon-Shor code has p th = [36]. The material presented in this Chapter has been published in[1]. 17

36 2 The Planar Code data qubit z z z z z z z x z x x x ancilla qubits (a) Toric code qubit arrangement shown in 3D. (b) Equivalent lattice in 2D with periodic boundaries. Figure 2.1: Arrangement of qubits on a torus with nearest-neighbour interactions. The stabiliser generators are the tensor products of Z and X on the qubits around faces and intersections respectively. 2.1 The Toric and Planar Codes Consider a collection of qubits arranged on the surface of a torus as shown in Fig. 2.1a, or more easily represented as a regular 2D array of qubits with periodic boundary conditions (Fig. 2.1b). The qubits are divided into two categories: data qubits located on the lines of the grid, and ancilla qubits on the faces and the intersections. The stabiliser generators for the toric code are the tensor products of Z on the four data qubits around each face, and the tensor products of X on the four data qubits around each intersection. Neighbouring stabilisers share two data qubits ensuring that adjacent X and Z stabilisers commute. Using the ancilla qubits, the eigenvalues of these stabilisers may be measured whilst still preserving a quantum state. We will always assume that the computer is initialised to the simultaneous +1 eigenstate of every stabiliser. An X error on a data qubit anti-commutes with the two adjacent Z stabilisers. Assuming no other errors occur, we observe a change in the measured eigenvalue of the adjacent Z stabilisers, from +1 to 1. Similarly, Z errors result in changes in the measured eigenvalues of two adjacent X stabilisers. In general, if many closely separated errors occur, one observes the terminals of chains of errors (Fig. 2.2). The interspersed ancilla allow for these eigenvalues to be measured using only local interactions. The configuration of eigenvalues measured by the ancilla on the faces forms the X syndrome from which X errors may be corrected. Similarly, Z errors are corrected using syndrome 18

37 2.1 The Toric and Planar Codes -1 (a) -1-1 (b) (c) Figure 2.2: Examples of error syndromes on the toric and planar codes. The state is initialised to the +1 eigenstate of all stabilisers. Shaded qubits indicate locations of X errors. (a) A single X error toggles the eigenvalues measured in the adjacent faces in the next syndrome extraction cycle. A Z error would be detected by the neighbouring intersections. (b) When errors are closely spaced, one observes only the terminals of a continuous chain. (c) Error chains can wrap around the boundaries in the toric code, but not in the planar code. information from ancilla located on the intersections. Logical operations can be associated with the four non-homotopic closed paths around the torus which cannot be contracted to a single stabiliser generator, shown in Fig (Two paths are homotopic if one can be continuously deformed into the other, in this case by multiplying by stabilisers). This particular topology defines two logical qubits: one from X (1) and Z (1), the other from X (2) and Z (2). These closed rings always overlap at an odd number of sites, yielding the correct commutation relations between logical operations. The distance d of a code is the length of the shortest logical operation. X X X X X Z Z Z Z Z Z Z Z Z Z X X X X X (a) X (1) and Z (1). (b) X (2) and Z (2). Figure 2.3: The two logical-x and logical-z operators in the toric code. Other rings of single-qubit X or Z operators on the lattice can either be deformed to products of the above rings, or contracted down to a single stabiliser generator. 19