Topological quantum error-correction codes are extremely practical, typically requiring only a two-dimensional lattice of qubits with tunable nearest-neighbor interactions yet tolerating high physical error rates $p$. It is computationally expensive to simulate the performance of such codes at low $p$, yet this is a regime we wish to study as low physical error rates lead to low qubit overhead. We present a very general method of analytically estimating the low-$p$ performance of the most promising class of topological codes. Our method can handle arbitrary periodic quantum circuits implementing the error detection associated with this class of codes, and arbitrary Pauli error models for each type of quantum gate. Our analytic expressions take only seconds to obtain, versus hundreds of hours to perform equivalent low-$p$ simulations.
Topological quantum error-correction codes are extremely practical, typically requiring only a two-dimensional lattice of qubits with tunable nearest-neighbor interactions yet tolerating high physical error rates p. It is computationally expensive to simulate the performance of such codes at low p, yet this is a regime we wish to study as low physical error rates lead to low qubit overhead. We present a very general method of analytically estimating the low-p performance of the most promising class of topological codes. Our method can handle arbitrary periodic quantum circuits implementing the error detection associated with this class of codes, and arbitrary Pauli error models for each type of quantum gate. Our analytic expressions take only seconds to obtain, versus hundreds of hours to perform equivalent low-p simulations.
In this thesis we present three main contributions to the field of topological quantum error correcting codes. We focus on some of the properties of such codes required for fault-tolerant quantum computation. Prior work has concentrated on determining error rate thresholds of particular models, but increasingly other parameters are gaining prominence. One of these is the overhead -- the quantity of a named resource required to achieve a desired level of performance from the code. We characterise the qubit overhead of the toric code in a fault-tolerant setting. These results provide a general framework for determining the overhead for other code constructions with more complicated noise models. Next we introduce a decoding algorithm, applicable to topological codes in a qudit architecture, specifically those where fault-tolerance is achieved through repeated syndrome measurements. It is computationally light and capable of decoding qudits of arbitrarily high dimension with negligible increase in its run time. The threshold of the decoder is limited by the percolation of the syndromes. Using local matching techniques we are able to overcome this limitation, increasing the threshold by almost a factor of two for high qudit dimensions. Finally, we turn our attention to a second family of topological quantum codes: the colour codes. In three and higher spatial dimensions such codes can support transversal non-Clifford gates. We show, using a technique that we call a star-bipartition of the vertices of the lattice, that any existing qubit colour code lattice can be used to define a qudit colour code. By generalising the notion of triorthogonal matrices we derive analogous transversality properties in the qudit codes.
This work compares the overhead of quantum error correction with concatenated and topological quantum error-correcting codes. To perform a numerical analysis, we use the Quantum Resource Estimator Toolbox (QuRE) that we recently developed. We use QuRE to estimate the number of qubits, quantum gates, and amount of time needed to factor a 1024-bit number on several candidate quantum technologies that differ in their clock speed and reliability. We make several interesting observations. First, topological quantum error correction requires fewer resources when physical gate error rates are high, white concatenated codes have smaller overhead for physical gate error rates below approximately 10E-7. Consequently, we show that different error-correcting codes should be chosen for two of the studied physical quantum technologies - ion traps and superconducting qubits. Second, we observe that the composition of the elementary gate types occurring in a typical logical circuit, a fault-tolerant circuit protected by the surface code, and a fault-tolerant circuit protected by a concatenated code all differ. This also suggests that choosing the most appropriate error correction technique depends on the ability of the future technology to perform specific gates efficiently.
Understanding the performance of realistic noisy encoded circuits is an important task for the development of large-scale practical quantum computers. Specifically, the development of proposals for quantum computation must be well informed by both the qualities of the low-level physical system of choice, and the properties of the high-level quantum error correction and fault-tolerance schemes. Gaining insight into how a particular computation will play out on a physical system is in general a difficult problem, as the classical simulation of arbitrary noisy quantum circuits is inefficient. Nevertheless, important classes of noisy circuits can be simulated efficiently. Such simulations have led to numerical estimates of threshold errors rates and resource estimates in topological codes subject to efficiently simulable error models. \n \nThis thesis describes and analyzes a method that my collaborators and I have introduced for leveraging efficient simulation techniques to understand the performance of large quantum processors that are subject to errors lying outside of the efficient simulation algorithm's applicability. The idea is to approximate an arbitrary gate error with an error from the efficiently simulable set in a way that ``honestly'' represents the original error's ability to preserve or distort quantum information. After introducing and analyzing the individual gate approximation method, its utility as a means for estimating circuit performance is studied. In particular, the method is tested within the use-case for which it was originally conceived; understanding the performance of a hypothetical physical implementation of a quantum error-correction protocol. It is found that the method performs exactly as desired in all cases. That is, the circuits composed of the approximated error models honestly represent the circuits composed of the errors derived from the physical models.
Utility-scale quantum computers require quantum error correcting codes with large numbers of physical qubits to achieve sufficiently low logical error rates. The performance of quantum error correction (QEC) is generally predicted through large-scale numerical simulations, used to estimate thresholds, finite-size scaling, and exponential suppression of logical errors below threshold. The connection of QEC to models from statistical mechanics provides an alternative tool for analysing QEC performance. However, predicting the behaviour of these models also requires large-scale numerical simulations, as analytic solutions are not generally known. Here we exploit an exact mapping, from a toric code under bit-flip noise that is post-selected on being syndrome free to the exactly-solvable two-dimensional Ising model on a torus, to derive an analytic solution for the logical failure rate across its full domain of physical error rates. In particular, this mapping provides closed-form expressions for the logical failure rate in four distinct regimes: the path-counting, below-threshold (ordered), near-threshold (critical), and above-threshold (disordered) regimes. Our framework places a number of familiar and long-standing numerical observations on firm theoretical ground. It also motivates explicit ansatze for the conventional QEC setting of non-post-selected codes whose statistical mechanics mappings involve random-bond disorder. Specifically, we introduce an effective tension model for the below-threshold regime, and a new scaling ansatz for the near-threshold regime, derived from an analysis of the domain wall energy cost distributions. By bridging statistical mechanics theory and quantum error correction practice, our results offer a new toolkit for designing, benchmarking, and understanding topological codes beyond current computational limits.
Topological quantum codes are intrinsically fault-tolerant to local noise, and underlie the theory of topological phases of matter. We explore geometry to enhance the performance of topological quantum codes by rotating the four dimensional self-correcting quantum memory, and present codes targeted to both near-term and utility-scale quantum computers. We identify a full set of logical Clifford operations and with it design a universal fault-tolerant quantum architecture. Our design achieves single-shot error correction, significant reductions in required qubits, and low-depth logical operations. In turn, our proposed architecture relaxes the requirements for achieving fault tolerance and offers an efficient path for realization in several near-term quantum hardware implementations. Our [[96,6,8]] 4D Hadamard lattice code has low weight-6 stabilizers and depth-8 syndrome extraction circuits, a high pseudo-threshold of ∼0.01, and a logical error rate of ∼10−6 per logical qubit per round of error correction at 10−3 physical error rate under a standard circuit-level noise model. A Clifford-complete logical gate set is presented, including a constructive and efficient method for Clifford gate synthesis.
The employment of quantum error correction codes (QECCs) within quantum computers potentially offers a reliability improvement for both quantum computation and communications tasks. However, the laws of quantum mechanics prevent us from directly invoking the mature family of classical error correction codes in the quantum domain. In order to circumvent the associated problems, the notion of quantum stabilizer codes (QSCs) was proposed in conjunction with syndrome-based decoding in the quantum regime. However, most of the powerful QSC schemes require long codewords for achieving a high performance, which potentially imposes additional challenges for implementation concerning their implementation, since their decoding may require longer than the quantum circuit's coherence time. Hence at the time of writing, QSCs exhibiting short to moderate codeword lengths are preferable. We commence by describing the pivotal problem encountered by classical error-correction codes, which also emerges when designing the QSCs, namely the intrinsic trade-off between the minimum distance versus coding rate. The complete formulation of this particular trade-off does not exist, but several lower and upper bounds can be found in the literature. It has been shown that a substantial gap can be observed between the upper and lower bound of the minimum distance, given the codeword length and the quantum coding rate. Hence, we propose an appealingly simple and invertible analytical approximation, for characterizing the trade-off between the quantum coding rate and the minimum distance of QSCs as well as their corresponding quantum bit error rate (QBER) performance upper-bound. For example, for a half-rate QSC having a codeword length of n = 128, the minimum distance is bounded by 11 < d < 22, while our approximation yields a minimum distance of d = 17 for the above-mentioned code. Next, we link this parametric study of the minimum distance versus quantum coding rate to the popular QSCs, namely to the family of quantum topological error correction codes (QTECCs). In order to construct the classical-to-quantum isomorphism, we conceive and investigate the family of classical topological error correction codes (TECCs), assuming that the bits of a codeword can be arranged in a lattice structure. We then present the classical-to-quantum isomorphism to pave the way for constructing their dual pairs in the quantum domain, which are the QTECCs. Finally, we characterize the performance of QTECCs in the face of the quantum depolarizing channel in terms of both their QBER and fidelity. Specifically, we demonstrate that for quantum coding rate rQ ~ 0, the threshold probability of the QBER below which the colour, rotated-surface, surface, and toric codes become capable of improving the uncoded QBER are given by 1.8 ×10-2, 1.3 ×10-2, 6.3 ×10-2 and 6.8 ×10-2, respectively. Furthermore, we also demonstrate that we can achieve beneficial fidelity improvements above the minimum fidelity of 0.94, 0.97 and 0.99 by employing the rQ = 1/7 colour code, the rQ = 1/9 rotated-surface code, and the rQ = 1/13 surface code, respectively. However, QSCs require additional quantum gates for their employment. Incorporating more quantum gates for performing error correction potentially introduces further sources of quantum decoherence into quantum computers. In this scenario, the primary challenge is to find the sufficient condition required by each of the quantum gates for beneficially employing QECCs in order to yield reliability improvements, given that the quantum gates utilized by the QECCs also introduce quantum decoherence. In this treatise, we approach this problem by firstly presenting the general framework of protecting quantum gates by the amalgamation of the transversal configuration of quantum gates and QSCs, which can be viewed as syndrome-based QECCs. Secondly, we provide examples of the advocated framework by invoking QTECCs for protecting both transversal Hadamard gates and controlled (CNOT) gates. Both our simulation and analytical results explicitly show that by utilizing QTECCs, the fidelity of the quantum gates can be beneficially improved, provided that quantum gates satisfying a certain minimum depolarization fidelity threshold (Fth) are available. For instance, for protecting transversal Hadamard gates, the minimum fidelity values required for each of the gates in order to attain fidelity improvements are 99.74%, 99.73%, 99.87%, and 99.86%, when they are protected by colour, rotated-surface, surface, and toric codes, respectively. Unfortunately, these specific Fth values can only be obtained for a very large number of physical qubits (n → ∞), when the quantum coding rate of the QTECCs approaches zero (rQ → 0). Finally, in order to conceive QSCs exhibiting a high quantum coding rate, we modify the construction of QTECCs for conceiving a low-complexity concatenated quantum turbocode (QTC). The above-mentioned high quantum coding rate is obtained by combining the quantum-domain version of short-block codes (SBCs) also known as single parity-check (SPC) codes as the outer codes and quantum unity-rate codes (QURCs) as the inner codes. Despite its design simplicity, the proposed QTC yields a near-hashing-bound error correction performance. For instance, compared to the best half-rate QTC known in the literature, namely the quantum irregular convolutional codes (QIrCCs) combined with the QURC scheme, which operates at the distance of D = 0.037 from the quantum hashing bound, our novel QSBC-QURC scheme can operate at the lower distance of D = 0:029. It is also worth mentioning that this is the first instantiation of QTCs capable of adjusting the quantum encoders according to the quantum coding rate required for mitigating the Pauli errors imposed by the time-variant depolarizing probabilities of the quantum channel.