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Quantum Error Correction with Biased Noise

Peter Brooks

Published May 31, 2013• 7 citations
Publisher

Abstract

Quantum computing offers powerful new techniques for speeding up the calculation of many classically intractable problems. Quantum algorithms can allow for the efficient simulation of physical systems, with applications to basic research, chemical modeling, and drug discovery; other algorithms have important implications for cryptography and internet security. At the same time, building a quantum computer is a daunting task, requiring the coherent manipulation of systems with many quantum degrees of freedom while preventing environmental noise from interacting too strongly with the system. Fortunately, we know that, under reasonable assumptions, we can use the techniques of quantum error correction and fault tolerance to achieve an arbitrary reduction in the noise level. In this thesis, we look at how additional information about the structure of noise, or "noise bias," can improve or alter the performance of techniques in quantum error correction and fault tolerance. In Chapter 2, we explore the possibility of designing certain quantum gates to be extremely robust with respect to errors in their operation. This naturally leads to structured noise where certain gates can be implemented in a protected manner, allowing the user to focus their protection on the noisier unprotected operations. In Chapter 3, we examine how to tailor error-correcting codes and fault-tolerant quantum circuits in the presence of dephasing biased noise, where dephasing errors are far more common than bit-flip errors. By using an appropriately asymmetric code, we demonstrate the ability to improve the amount of error reduction and decrease the physical resources required for error correction. In Chapter 4, we analyze a variety of protocols for distilling magic states, which enable universal quantum computation, in the presence of faulty Clifford operations. Here again there is a hierarchy of noise levels, with a fixed error rate for faulty gates, and a second rate for errors in the distilled states which decreases as the states are distilled to better quality. The interplay of of these different rates sets limits on the achievable distillation and how quickly states converge to that limit.

DOI
10.7907/1tvt-j780

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Quantum error correction and fault-tolerance have provided the possibility for large scale quantum computations without a detrimental loss of quantum information. A very natural class of gates for fault-tolerant quantum computation is the Clifford gate set and as such their usefulness for universal quantum computation is of great interest. Clifford group gates augmented by magic state preparation give the possibility of simulating universal quantum computation. However, experimentally one cannot expect to perfectly prepare magic states. Nonetheless, it has been shown that by repeatedly applying operations from the Clifford group and measurements in the Pauli basis, the fidelity of noisy prepared magic states can be increased arbitrarily close to a pure magic state [1]. We investigate the robustness of magic state distillation to perturbations of the initial states to arbitrary locations in the Bloch sphere due to noise. Additionally, we consider a depolarizing noise model on the quantum gates in the decoding section of the distillation protocol and demonstrate its effect on the convergence rate and threshold value. Finally, we establish that faulty magic state distillation is more efficient than fault-tolerance-assisted magic state distillation at low error rates due to the large overhead in the number of quantum gates and qubits required in a fault-tolerance architecture. The ability to perform magic state distillation with noisy gates leads us to conclude that this could be a realistic scheme for future small-scale quantum computing devices as fault-tolerance need only be used in the final steps of the protocol.

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•Jun 1, 2020•2 citations
Quantum computing offers powerful new techniques for speeding up the calculation of many classically intractable problems. Quantum algorithms can allow for the efficient simulation of physical systems, with applications to basic research, chemical modeling, and drug discovery; other algorithms have important implications for cryptography and internet security. At the same time, building a quantum computer is a daunting task, requiring the coherent manipulation of systems with many quantum degrees of freedom while preventing environmental noise from interacting too strongly with the system. Fortunately, we know that, under reasonable assumptions, we can use the techniques of quantum error correction and fault tolerance to achieve an arbitrary reduction in the noise level. In this thesis, we look at how additional information about the structure of noise, or "noise bias," can improve or alter the performance of techniques in quantum error correction and fault tolerance. In Chapter 2, we explore the possibility of designing certain quantum gates to be extremely robust with respect to errors in their operation. This naturally leads to structured noise where certain gates can be implemented in a protected manner, allowing the user to focus their protection on the noisier unprotected operations. In Chapter 3, we examine how to tailor error-correcting codes and fault-tolerant quantum circuits in the presence of dephasing biased noise, where dephasing errors are far more common than bit-flip errors. By using an appropriately asymmetric code, we demonstrate the ability to improve the amount of error reduction and decrease the physical resources required for error correction. In Chapter 4, we analyze a variety of protocols for distilling magic states, which enable universal quantum computation, in the presence of faulty Clifford operations. Here again there is a hierarchy of noise levels, with a fixed error rate for faulty gates, and a second rate for errors in the distilled states which decreases as the states are distilled to better quality. The interplay of of these different rates sets limits on the achievable distillation and how quickly states converge to that limit.
Quantum computers have the potential to solve several interesting problems in polynomial \ntime for which no polynomial time classical algorithms have been found. However, \none of the major challenges in building quantum devices is that quantum systems are very \nsensitive to noise arising from undesired interactions with the environment. Noise can lead \nto errors which can corrupt the results of the computation. Quantum error correction is \none way to mitigate the effects of noise arising in quantum devices. \nWith a plethora of quantum error correcting codes that can be used in various settings, \none of the main challenges of quantum error correction is understanding how well various \ncodes perform under more realistic noise models that can be observed in experiments. \nThis thesis proposes a new decoding algorithm which can optimize threshold values of \nerror correcting codes under different noise models. The algorithm can be applied to \nany Markovian noise model. Further, it is shown that for certain noise models, logical \nClifford corrections can further improve a code's threshold value if the code obeys certain \nsymmetries. \n \nSince gates and measurements cannot in general be performed with perfect precision, \nthe operations required to perform quantum error correction can introduce more errors \ninto the system thus negating the benefits of error correction. Fault-tolerant quantum \ncomputing is a way to perform quantum error correction with imperfect operations while \nretaining the ability to suppress errors as long as the noise is below a code's threshold. \nOne of the main challenges in performing fault-tolerant error correction is the high resource \nrequirements that are needed to obtain very low logical noise rates. With the use of \nflag qubits, this thesis develops new fault-tolerant error correction protocols that are applicable \nto arbitrary distance codes. Various code families are shown to satisfy the requirements \nof flag fault-tolerant error correction. We also provide circuits using a constant number of \nqubits for these codes. It is shown that the proposed flag fault-tolerant method uses fewer \nqubits than previous fault-tolerant error correction protocols. \n \nIt is often the case that the noise afflicting a quantum device cannot be fully characterized. \nFurther, even with some knowledge of the noise, it can be very challenging to use \nanalytic decoding methods to improve the performance of a fault-tolerant scheme. This \nthesis presents decoding schemes using several state of the art machine learning techniques \nwith a focus on fault-tolerant quantum error correction in regimes that are relevant to near \nterm experiments. It is shown that even in low noise rate regimes and with no knowledge \nof the noise, noise can be further suppressed for small distance codes. Limitations of machine \nlearning decoders as well as the classical resources required to perform active error \ncorrection are discussed. \n \nIn many cases, gate times can be much shorter than typical measurement times of \nquantum states. Further, classical decoding of the syndrome information used in quantum \nerror correction to compute recovery operators can also be much slower than gate times. \nFor these reasons, schemes where error correction can be implemented in a frame (known \nas the Pauli frame) have been developed to avoid active error correction. In this thesis, we \ngeneralize previous Pauli frame schemes and show how Clifford frame error correction can \nbe implemented with minimal overhead. Clifford frame error correction is necessary if the \nlogical component of recovery operators were chosen from the Clifford group, but could \nalso be used in randomized benchmarking schemes.
Qubits with strongly biased noise, in which phase-flip errors are orders of magnitude more frequent than bit-flips, arise both naturally, as in electron and nuclear spins, and by engineering, as in stabilized cat qubits. This noise structure holds the promise of reducing the daunting hardware overhead of fault-tolerant quantum computing, but exploiting it requires physical operations that do not convert frequent phase-flips into rare bit-flips. In this review, we analyze the most prominent fault-tolerant protocols for biased-noise qubits, organized according to the available set of such bias-preserving operations. When this set is restricted to the CZ gate together with preparation and measurement in the X basis, we show that the complexity of the required syndrome extraction gadgets essentially cancels the benefit of the noise bias: at experimentally relevant error rates, one may as well ignore the bias and rely on standard error correction designed for depolarizing noise. The situation changes drastically when a bias-preserving CX gate is available: the hierarchy of errors can then be reflected in the structure of the code, with frequent phase-flips corrected by a dedicated high-threshold code and rare bit-flips by concatenation with a high-rate code. The same hierarchy also enables hardware-efficient preparation of magic states. Finally, as a bias-preserving CX is forbidden in naturally biased platforms and challenging in engineered ones, we present a measurement-based architecture in which a high-fidelity quantum non-demolition readout of multi-qubit Pauli Z operators takes its place, extending these overhead reductions to a much broader range of physical platforms.
Quantum error correction and fault-tolerance have provided the possibility for large scale quantum computations without a detrimental loss of quantum information. A very natural class of gates for fault-tolerant quantum computation is the Clifford gate set and as such their usefulness for universal quantum computation is of great interest. Clifford group gates augmented by magic state preparation give the possibility of simulating universal quantum computation. However, experimentally one cannot expect to perfectly prepare magic states. Nonetheless, it has been shown that by repeatedly applying operations from the Clifford group and measurements in the Pauli basis, the fidelity of noisy prepared magic states can be increased arbitrarily close to a pure magic state~\cite{Bravyi}. We investigate the robustness of magic state distillation to perturbations of the initial states to arbitrary locations in the Bloch sphere due to noise. Additionally, we consider a depolarizing noise model on the quantum gates in the decoding section of the distillation protocol and demonstrate its effect on the convergence rate and threshold value. Finally, we establish that faulty magic state distillation is more efficient than fault-tolerance-assisted magic state distillation at low error rates due to the large overhead in the number of quantum gates and qubits required in a fault-tolerance architecture. The ability to perform magic state distillation with noisy gates leads us to conclude that this could be a realistic scheme for future small-scale quantum computing devices as fault-tolerance need only be used in the final steps of the protocol.
Quantum error correction and fault-tolerance have provided the possibility for large scale quantum computations without a detrimental loss of quantum information. A very natural class of gates for fault-tolerant quantum computation is the Clifford gate set and as such their usefulness for universal quantum computation is of great interest. Clifford group gates augmented by magic state preparation give the possibility of simulating universal quantum computation. However, experimentally one cannot expect to perfectly prepare magic states. Nonetheless, it has been shown that by repeatedly applying operations from the Clifford group and measurements in the Pauli basis, the fidelity of noisy prepared magic states can be increased arbitrarily close to a pure magic state [1]. We investigate the robustness of magic state distillation to perturbations of the initial states to arbitrary locations in the Bloch sphere due to noise. Additionally, we consider a depolarizing noise model on the quantum gates in the decoding section of the distillation protocol and demonstrate its effect on the convergence rate and threshold value. Finally, we establish that faulty magic state distillation is more efficient than fault-tolerance-assisted magic state distillation at low error rates due to the large overhead in the number of quantum gates and qubits required in a fault-tolerance architecture. The ability to perform magic state distillation with noisy gates leads us to conclude that this could be a realistic scheme for future small-scale quantum computing devices as fault-tolerance need only be used in the final steps of the protocol.
Quantum error correction is the backbone of fault-tolerant quantum computation, a necessary requirement for any large scale quantum computer. The fault-tolerance threshold theorem has long been a target for experimental precision, allowing for the possibility of reducing logical error rates to arbitrarily low levels without excessive overhead. While there are many promising fault-tolerant architectures, the path towards the most practical fault-tolerant scheme is far from decided and may vary for differing physical models. This thesis proposes new schemes for universal fault-tolerant quantum computation in both the concatenated and topological code settings. \n \nThrough the concatenation of two different error correcting codes, a set of universal fault-tolerant gates can be obtained without requiring the need for magic state distillation. A lower bound of 1.28times10−31.28 \\times 10^{-3}1.28times10−3 for the fault-tolerance threshold under circuit level depolarizing noise is obtained. Additionally, stacked codes are proposed as a means to simulate the action of a 3D topological code in 2D, allowing for the application of a universal set of transversal operations. While fault-tolerant, unfortunately the scheme does not exhibit a threshold due to the decreasing pseudo-threshold with growing code distance, yet points to potential interesting avenues for fault-tolerant computation in 2D without distillation. \n \nOne of the primary avenues to constructing fault-tolerant logical operations is through transversal operations. In this thesis, the set of single qubit logical gates that can be implemented transversally are characterized and determined to all belong to the Clifford hierarchy. Moreover, any diagonal two-qubit operation that can be applied transversally must belong to the same level of the Clifford hierarchy as the set of gates that can be implemented in the single-qubit case. \n \nThe opposite to quantum error correction is privacy, where the output of a channel is disguised from its input. The two are fundamentally related through the complementary channel construction. This thesis presents a new class of private quantum channels, expanding the existing class beyond a seemingly fundamental restriction. This yields interesting insights into the structure of quantum information and the leaking of information to external environments. Additionally, the duality is only recovered when extending the complementary channel to sufficiently high environmental dimension. \n \nFinally, the error properties of bucket brigade quantum Random Access Memory (qRAM) are assessed. It is determined that using the bucket brigade qRAM architecture for the running of Grover's algorithm necessitates reducing the error rate of the individual components to exponentially small levels for an exponential sized memory. As such, fault-tolerant architectures will likely play an essential role in the construction of such computing primitives.
This thesis is concerned with fault-tolerant quantum information processing using quantum error-correcting codes. It contains two major pieces of work. The first is a study of coherent noise in the context of stabilizer error-correcting codes. The second is a proposed scheme for a universal set of fault-tolerant logical gates in a particular code family built out of the 3D toric code. Chapter 1 provides an introduction to quantum computation and fault tolerance. Many basic concepts in error-correcting codes are defined. Special attention is paid to the set of code properties that are most likely to determine how easily a given fault-tolerant scheme might be implemented on a physical device. These include the fault-tolerant noise threshold and the overhead. In Chapters 2 and 3 we study the effectiveness of quantum error correction against coherent noise. Coherent errors (for example, unitary noise) can interfere constructively, so that in some cases the average infidelity of a quantum circuit subjected to coherent errors may increase quadratically with the circuit size; in contrast, when errors are incoherent (for example, depolarizing noise), the average infidelity increases at worst linearly with circuit size. We consider the performance of quantum stabilizer codes against a noise model in which a unitary rotation is applied to each qubit, where the axes and angles of rotation are nearly the same for all qubits. In Chapter 2 we introduce coherent noise and incoherent noise and a number of methods that are useful for the study of coherent noise. We study the repetition code as a basic example, and we also study a correlated noise model. In Chapter 3 we show that for the toric code subject to such independent coherent noise, and for minimal-weight decoding, the logical channel after error correction becomes increasingly incoherent as the length of the code increases, provided the noise strength decays inversely with the code distance. A similar conclusion holds for weakly correlated coherent noise. Our methods can also be used for analyzing the performance of other codes and fault-tolerant protocols against coherent noise. However, our result does not show that the coherence of the logical channel is suppressed in the more physically relevant case where the noise strength is held constant as the code block grows, and we recount the difficulties that prevented us from extending the result to that case. Nevertheless our work supports the idea that fault-tolerant quantum computing schemes will work effectively against coherent noise, providing encouraging news for quantum hardware builders who worry about the damaging effects of control errors and coherent interactions with the environment. Chapter 4 is connected to another aspect of fault tolerance, fault-tolerant logical gates. The toric code is a promising candidate for fault-tolerant quantum computation because of its high threshold and low-weight stabilizers. A universal gate set in the toric code generally requires magic state distillation, which can incur a significant qubit overhead. In this work we construct an error-correcting code in three dimensions based on the toric code that features a fault-tolerant T gate with no magic state distillation required. We further describe a subsystem version of our code which supports a universal set of fault-tolerant gates. This code can be converted into the stabilizer version using gauge-fixing. We also argue that our code can be converted to a (2+1)-D protocol, where a 2D lattice undergoes a measurement-based protocol over time. In this way, a fault-tolerant logical T gate can be realized in a 2D toric code structure.