Fault-Tolerant Quantum Computation

Fault-Tolerant Postselected Quantum Computation: Threshold Analysis

The schemes for fault-tolerant postselected quantum computation given in [Knill, Fault-Tolerant Postselected Quantum Computation: Schemes, http://arxiv.org/abs/quant-ph/0402171] are analyzed to determine their error-tolerance. The analysis is based on computer-assisted heuristics. It indicates that if classical and quantum communication delays are negligible, then scalable qubit-based quantum computation is possible with errors above 1% per elementary quantum gate.

Fault-Tolerant Exact State Transmission

We show that a category of one-dimensional XY-type models may enable high-fidelity quantum state transmissions, regardless of details of coupling configurations. This observation leads to a fault- tolerant design of a state transmission setup. The setup is fault-tolerant, with specified thresholds, against engineering failures of coupling configurations, fabrication imperfections or defects, and even time-dependent noises. We propose the implementation of the fault-tolerant scheme using hard-core bosons in one-dimensional optical lattices.

Fault-tolerant conversion between the Steane and Reed-Muller quantum codes

Steane's 7-qubit quantum error-correcting code admits a set of fault-tolerant gates that generate the Clifford group, which in itself is not universal for quantum computation. The 15-qubit Reed-Muller code also does not admit a universal fault-tolerant gate set but possesses fault-tolerant T and control-control-Z gates. Combined with the Clifford group, either of these two gates generate a universal set. Here, we combine these two features by demonstrating how to fault-tolerantly convert between these two codes, providing a new method to realize universal fault-tolerant quantum computation. On…

Opportunities and Challenges in Fault-Tolerant Quantum Computation

I will give an overview of what I see as some of the most important future directions in the theory of fault-tolerant quantum computation. In particular, I will give a brief summary of the major problems that need to be solved in fault tolerance based on low-density parity check codes and in hardware-specific fault tolerance. I will then conclude with a discussion of a possible new paradigm for designing fault-tolerant protocols based on a space-time picture of quantum circuits.

A Small Quantum Computer is Needed to Optimize Fault-Tolerant Protocols

As far as we know, a useful quantum computer will require fault-tolerant gates, and existing schemes demand a prohibitively large space and time overhead. We argue that a first generation quantum computer will be very valuable to design, test, and optimize fault-tolerant protocols tailored to the noise processes of the hardware. Our argument is essentially a critical analysis of the current methods envisioned to optimize fault-tolerant schemes, which rely on hardware characterization, noise modelling, and numerical simulations. We show that, even within a very restricted set of noise models, e…

Constructing arbitrary Steane code single logical qubit fault-tolerant gates

We present a simple method for constructing optimal fault-tolerant approximations of arbitrary unitary gates using an arbitrary discrete universal gate set. The method presented is numerical and scales exponentially with the number of gates used in the approximation. However, for the specific case of arbitrary single-qubit gates and the fault-tolerant gates permitted by the concatenated 7-qubit Steane code, we find gate sequences sufficiently long and accurate to permit the fault-tolerant factoring of numbers thousands of bits long. A general scaling law of how rapidly these fault-tolerant app…

Simple fault-tolerant encoding over q-ary CSS quantum codes

CSS codes are a subfamily of stabilizer codes especially appropriate for fault-tolerant quantum computations. A very simple method is proposed to encode a general qudit when a Calderbank-Shor-Steane quantum code, defined over a q-ary alphabet, is used.

Fault-tolerant holonomic quantum computation

We explain how to combine holonomic quantum computation (HQC) with fault tolerant quantum error correction. This establishes the scalability of HQC, putting it on equal footing with other models of computation, while retaining the inherent robustness the method derives from its geometric nature.

Threshold Estimate for Fault Tolerant Quantum Computation

I make a rough estimate of the accuracy threshold for fault tolerant quantum computing with concatenated codes. First I consider only gate errors and use the depolarizing channel error model. I will follow P.Shor (quant-ph/9505011) for fault tolerant error correction (FTEC) and the fault tolerant implementation of elementary operations on states encoded by the 7-qubit code. A simple computer simulation suggests a threshold for gate errors of the order ε\approx 10^{-3} or better. I also give a simple argument that the threshold for memory errors is about 10 times smaller, thus ε\approx 10^{-4}.

灵活的容错门构件

Flexible Fault Tolerant Gate Gadgets

本研究设计出可让数据与辅助比特采用不同编码的灵活容错门构件，通过为辅助比特选用稳定子码，实现了通用量子码上的Clifford与非Clifford门容错操作，首次提出非加性码的通用容错门集合，并验证了其在两类信道下的容错性。

Deriving a Fault-Tolerant Threshold for a Global Control Scheme

In this paper, adapted from the author's PhD thesis, we present otherwise unpublished results relating to global control schemes, culminating in the calculation of a fault-tolerant threshold for one such scheme. As with early fault-tolerant threshold results, the aim is to calculate a positive number, not to optimise it. We also discuss how the results might affect other related schemes, such as those based on cellular automata.

Fibonacci schemes of fault-tolerant quantum computation

The threshold estimate derived in previous versions of this paper was incorrect; this note explains the flaw. A new proof is discussed in arXiv:0809.5063.

环面码上的容错常深度Clifford门

Fault-Tolerant Constant-Depth Clifford Gates on Toric Codes

针对2D环面码，提出并模拟了结合折叠横向门、Dehn扭转和单次逻辑Pauli测量的容错常深度逻辑门组，其能生成完整Clifford群并具备容错性与常深度特性。

Fault-tolerant gates via homological product codes

A method for the implementation of a universal set of fault-tolerant logical gates is presented using homological product codes. In particular, it is shown that one can fault-tolerantly map between different encoded representations of a given logical state, enabling the application of different classes of transversal gates belonging to the underlying quantum codes. This allows for the circumvention of no-go results pertaining to universal sets of transversal gates and provides a general scheme for fault-tolerant computation while keeping the stabilizer generators of the code sparse.

Fault-Tolerant Topological One-Way Quantum Computation with Probabilistic Two-Qubit Gates

We propose a scalable way to construct a 3D cluster state for fault-tolerant topological one-way computation (TOWC) even if the entangling two-qubit gates succeed with a small probability. It is shown that fault-tolerant TOWC can be performed with the success probability of the two-qubit gate such as 0.5 (0.1) provided that the conditional error probability of the two-qubit gate is less than 0.040% (0.016%). Furthermore, the resource usage is considerably suppressed compared to the conventional fault-tolerant schemes with probabilistic two-qubit gates.

一种基于三维簇态的灵活的GKP态嵌入容错量子计算配置

A Flexible GKP-State-Embedded Fault-Tolerant Quantum Computation Configuration Based on a Three-Dimensional Cluster State

本文提出一种基于三维簇态的灵活GKP态嵌入容错量子计算架构，利用多种量子资源和更多自由度实现通用容错量子计算，通过设计光学纠缠生成器和时间延迟分束器网络构建三维簇态，并采用部分压缩表面GKP码实现容错计算，最终确定最佳压缩门参数达到11.5 dB的容错阈值。

Improved ancilla preparation scheme increases fault-tolerant threshold

We demonstrate an improved concatenated encoded ancilla preparation procedure. Simulations show that this procedure significantly increases the error threshold beneath which arbitrarily long quantum computations are possible.

一种紧凑的单向容错光学量子计算

A Compact One-Way Fault-Tolerant Optical Quantum Computation

针对三维簇态制备难题，本文提出利用光学参量振荡器空间光谱模式构建大规模规范立方簇态，并结合 GKP 态降低容错压缩阈值，实现紧凑的单向容错光学量子计算。

Fault-Tolerant Quantum Computation with Constant Overhead

What is the minimum number of extra qubits needed to perform a large fault-tolerant quantum circuit? Working in a common model of fault-tolerance, I show that in the asymptotic limit of large circuits, the ratio of physical qubits to logical qubits can be a constant. The construction makes use of quantum low-density parity check codes, and the asymptotic overhead of the protocol is equal to that of the family of quantum error-correcting codes underlying the fault-tolerant protocol.

Fault-tolerant quantum error correction code conversion

In this paper we demonstrate how data encoded in a five-qubit quantum error correction code can be converted, fault-tolerantly, into a seven-qubit Steane code. This is achieved by progressing through a series of codes, each of which fault-tolerantly corrects at least one error. Throughout the conversion the encoded qubit remains protected. We found, through computational search, that the method used to convert between codes given in this paper is optimal.

Fault-Tolerant Quantum Computation for Local Leakage Faults

We provide a rigorous analysis of fault-tolerant quantum computation in the presence of local leakage faults. We show that one can systematically deal with leakage by using appropriate leakage-reduction units such as quantum teleportation. The leakage noise is described by a Hamiltonian and the noise is treated coherently, similar to general non-Markovian noise analyzed in Refs. quant-ph/0402104 and quant-ph/0504218. We describe ways to limit the use of leakage-reduction units while keeping the quantum circuits fault-tolerant and we also discuss how leakage reduction by teleportation is natura…

A Fault-Tolerant Scheme of Holonomic Quantum Computation on Stabilizer Codes with Robustness to Low-weight Thermal Noise

We show an equivalence relation between fault-tolerant circuits for a stabilizer code and fault-tolerant adiabatic processes for holonomic quantum computation (HQC), in the case where quantum information is encoded in the degenerated ground space of the system Hamiltonian. By this equivalence, we can systematically construct a fault-tolerant HQC scheme, which can geometrically implement a universal set of encoded quantum gates by adiabatically deforming the system Hamiltonian. During this process, quantum information is protected from low weight thermal excitations by an energy gap that does n…

Comment on "Resilient Quantum Computation in Correlated Environments: A Quantum Phase Transition Perspective" and "Fault-tolerant Quantum Computation with Longe-range Correlated Noise"

Critical comments on the recent papers supporting the idea of resilient quantum computations are presented.

Fault-Tolerant Operation of a Quantum Error-Correction Code

Quantum error correction protects fragile quantum information by encoding it into a larger quantum system. These extra degrees of freedom enable the detection and correction of errors, but also increase the operational complexity of the encoded logical qubit. Fault-tolerant circuits contain the spread of errors while operating the logical qubit, and are essential for realizing error suppression in practice. While fault-tolerant design works in principle, it has not previously been demonstrated in an error-corrected physical system with native noise characteristics. In this work, we experimenta…

Deutsch-Jozsa算法的容错实现

Fault-Tolerant Implementation of the Deutsch-Jozsa Algorithm

该研究利用$[[4,2,2]]$码在离子阱量子计算机上实现了Deutsch-Jozsa算法的容错版本，无需辅助比特，实验表明对所有预言机噪声平均降低近90%。

Scaling and renormalization in fault-tolerant quantum computers

This work is concerned with phrasing the concepts of fault-tolerant quantum computation within the framework of disordered systems, Bernoulli site percolation in particular. We show how the so-called "threshold theorems" on the possibility of fault-tolerant quantum computation with constant error rate can be cast as a renormalization (coarse-graining) of the site percolation process describing the occurrence of errors during computation. We also use percolation techniques to derive a trade-off between the complexity overhead of the fault-tolerant circuit and the threshold error rate.

Flag fault-tolerant error correction, measurement, and quantum computation for cyclic CSS codes

Flag qubits have recently been proposed in syndrome extraction circuits to detect high-weight errors arising from fewer faults. The use of flag qubits allows the construction of fault-tolerant protocols with the fewest number of ancillas known to-date. In this work, we prove some critical properties of CSS codes constructed from classical cyclic codes that enable the construction of a flag fault-tolerant error correction scheme. We then develop fault-tolerant protocols as well as a family of circuits for flag fault-tolerant error correction and operator measurement, requiring only four ancilla…

Fault-tolerant quantum computation with asymmetric Bacon-Shor codes

We develop a scheme for fault-tolerant quantum computation based on asymmetric Bacon-Shor codes, which works effectively against highly biased noise dominated by dephasing. We find the optimal Bacon-Shor block size as a function of the noise strength and the noise bias, and estimate the logical error rate and overhead cost achieved by this optimal code. Our fault-tolerant gadgets, based on gate teleportation, are well suited for hardware platforms with geometrically local gates in two dimensions.

Fault tolerant quantum key distribution protocol with collective random unitary noise

We propose an easy implementable prepare-and-measure protocol for robust quantum key distribution with photon polarization. The protocol is fault tolerant against collective random unitary channel noise. The protocol does not need any collective quantum measurement or quantum memory. A security proof and a specific linear optical realization using spontaneous parametric down conversion are given.

Scalable fault-tolerant quantum computation in DFS blocks

We investigate how to concatenate different decoherence-free subspaces (DFSs) to realize scalable universal fault-tolerant quantum computation. Based on tunable $XXZ$ interactions, we present an architecture for scalable quantum computers which can fault-tolerantly perform universal quantum computation by manipulating only single type of parameter. By using the concept of interaction-free subspaces we eliminate the need to tune the couplings between logical qubits, which further reduces the technical difficulties for implementing quantum computation.

Optical demonstration of quantum fault-tolerant threshold

A major challenge in practical quantum computation is the ineludible errors caused by the interaction of quantum systems with their environment. Fault-tolerant schemes, in which logical qubits are encoded by several physical qubits, enable correct output of logical qubits under the presence of errors. However, strict requirements to encode qubits and operators render the implementation of a full fault-tolerant computation challenging even for the achievable noisy intermediate-scale quantum technology. Here, we experimentally demonstrate the existence of the threshold in a special fault-toleran…

常空间开销容错量子输入/输出和通信

Constant-space-overhead fault-tolerant quantum input/output and communication

本研究提出了一种基于量子汉明码拼接的容错量子通信方法，通过常空间开销实现更高的通信速率。

面向近期量子纠错的确定性容错态制备：基于布尔可满足性的自动综合

Deterministic Fault-Tolerant State Preparation for Near-Term Quantum Error Correction: Automatic Synthesis Using Boolean Satisfiability

本文提出利用布尔可满足性等经典电路设计技术，为当前及近期硬件可实现的一类量子码自动综合确定性、最优容错态制备电路，并通过电路级噪声模拟验证。

Fault-tolerant fermionic quantum computation based on color code

An important approach to the fault-tolerant quantum computation is protecting the logical information using the quantum error correction. Usually, the logical information is in the form of logical qubits, which are encoded in physical qubits using quantum error correction codes. Compared with the qubit quantum computation, the fermionic quantum computation has advantages in quantum simulations of fermionic systems, e.g. molecules. In this paper, we show that the fermionic quantum computation can be universal and fault-tolerant if we encode logical Majorana fermions in physical Majorana fermion…

使用 Azure Quantum 资源估算器评估容错量子计算性能

Using Azure Quantum Resource Estimator for Assessing Performance of Fault Tolerant Quantum Computation

本文介绍 Azure Quantum 及其开发套件中的资源估算工具，可自动评估容错量子计算所需的逻辑与物理资源，并以三种乘法算法为例进行验证。

Fault-Tolerant Quantum Computation with Higher-Dimensional Systems

Instead of a quantum computer where the fundamental units are 2-dimensional qubits, we can consider a quantum computer made up of d-dimensional systems. There is a straightforward generalization of the class of stabilizer codes to d-dimensional systems, and I will discuss the theory of fault-tolerant computation using such codes. I prove that universal fault-tolerant computation is possible with any higher-dimensional stabilizer code for prime d.

A Study on the Noise Threshold of Fault-tolerant Quantum Error Correction

Quantum circuits implementing fault-tolerant quantum error correction (QEC) for the three qubit bit-flip code and five-qubit code are studied. To describe the effect of noise, we apply a model based on a generalized effective Hamiltonian where the system-environment interactions are taken into account by including stochastic fluctuating terms in the system Hamiltonian. This noise model enables us to investigate the effect of noise in quantum circuits under realistic device conditions and avoid strong assumptions such as maximal parallelism and weak storage errors. Noise thresholds of the QEC c…

Fault-tolerant quantum computation with high threshold in two dimensions

We present a scheme of fault-tolerant quantum computation for a local architecture in two spatial dimensions. The error threshold is 0.75% for each source in an error model with preparation, gate, storage and measurement errors.

A fault-tolerant one-way quantum computer

We describe a fault-tolerant one-way quantum computer on cluster states in three dimensions. The presented scheme uses methods of topological error correction resulting from a link between cluster states and surface codes. The error threshold is 1.4% for local depolarizing error and 0.11% for each source in an error model with preparation-, gate-, storage- and measurement errors.

Fault-Tolerant Quantum Computation via Exchange interactions

Quantum computation can be performed by encoding logical qubits into the states of two or more physical qubits, and controlling a single effective exchange interaction and possibly a global magnetic field. This "encoded universality" paradigm offers potential simplifications in quantum computer design since it does away with the need to perform single-qubit rotations. Here we show that encoded universality schemes can be combined with quantum error correction. In particular, we show explicitly how to perform fault-tolerant leakage correction, thus overcoming the main obstacle to fault-tolerant…

容错量子计算的三层架构

A Three-Layer Architecture for Fault-Tolerant Quantum Computing

提出一种硬件无关的三层容错量子计算架构，以容错层为桥梁连接逻辑程序与硬件执行，实现统一抽象模型。

Experimental demonstration of fault-tolerant state preparation with superconducting qubits

Robust quantum computation requires encoding delicate quantum information into degrees of freedom that are hard for the environment to change. Quantum encodings have been demonstrated in many physical systems by observing and correcting storage errors, but applications require not just storing information; we must accurately compute even with faulty operations. The theory of fault-tolerant quantum computing illuminates a way forward by providing a foundation and collection of techniques for limiting the spread of errors. Here we implement one of the smallest quantum codes in a five-qubit super…

Universal Fault-Tolerant Quantum Computing with Stabiliser Codes

The quantum logic gates used in the design of a quantum computer should be both universal, meaning arbitrary quantum computations can be performed, and fault-tolerant, meaning the gates keep errors from cascading out of control. A number of no-go theorems constrain the ways in which a set of fault-tolerant logic gates can be universal. These theorems are very restrictive, and conventional wisdom holds that a universal fault-tolerant logic gate set cannot be implemented natively, requiring us to use costly distillation procedures for quantum computation. Here, we present a general framework for…

New Limits on Fault-Tolerant Quantum Computation

We show that quantum circuits cannot be made fault-tolerant against a depolarizing noise level of approximately 45%, thereby improving on a previous bound of 50% (due to Razborov). Our precise quantum circuit model enables perfect gates from the Clifford group (CNOT, Hadamard, S, X, Y, Z) and arbitrary additional one-qubit gates that are subject to that much depolarizing noise. We prove that this set of gates cannot be universal for arbitrary (even classical) computation, from which the upper bound on the noise threshold for fault-tolerant quantum computation follows.

Universal fault-tolerant quantum computation with only transversal gates and error correction

Transversal implementations of encoded unitary gates are highly desirable for fault-tolerant quantum computation. Though transversal gates alone cannot be computationally universal, they can be combined with specially distilled resource states in order to achieve universality. We show that "triorthogonal" stabilizer codes, introduced for state distillation by Bravyi and Haah [Phys. Rev. A 86 052329 (2012)], admit transversal implementation of the controlled-controlled-Z gate. We then construct a universal set of fault-tolerant gates without state distillation by using only transversal controll…

存在死比特时的容错Hastings-Haah码

Fault-Tolerant Hastings-Haah Codes in the Presence of Dead Qubits

本文针对存在死比特的情况，开发了Hastings-Haah Floquet码的容错协议。

Fault-tolerant quantum computation with few qubits

Reliable qubits are difficult to engineer, but standard fault-tolerance schemes use seven or more physical qubits to encode each logical qubit, with still more qubits required for error correction. The large overhead makes it hard to experiment with fault-tolerance schemes with multiple encoded qubits. The 15-qubit Hamming code protects seven encoded qubits to distance three. We give fault-tolerant procedures for applying arbitrary Clifford operations on these encoded qubits, using only two extra qubits, 17 total. In particular, individual encoded qubits within the code block can be targeted. …

A note on threshold theorem of fault-tolerant quantum computation

Error-correction process has to be carried out periodically to prevent accumulation of errors in fault-tolerant quantum computation. It is believed that the best choice to get maximum threshold value is carrying out an error-correction process after each quantum gate operation. Result of this note shows that the optimal error-correction period depends on the value of k which is the level number of concatenated quantum error-correction code.

Quantum fault tolerance in small experiments

I discuss a variety of issues relating to near-future experiments demonstrating fault-tolerant quantum computation. I describe a family of fault-tolerant quantum circuits that can be performed with 5 qubits arranged on a ring with nearest-neighbor interactions. I also present a criterion whereby we can say that an experiment has succeeded in demonstrating fault tolerance. Finally, I discuss the possibility of using future fault-tolerant experiments to answer important questions about the interaction of fault-tolerant protocols with real experimental errors.

容错复形

Fault-tolerant complexes

本文提出融合复形与更一般的容错复形，用三维几何和同调语言统一描述表面码、颜色码及子系统码的容错协议，并自动发现627种方案及更优阈值。

Enhanced fault-tolerant quantum computing in $d$-level systems

Error correcting codes protect quantum information and form the basis of fault tolerant quantum computing. Leading proposals for fault-tolerant quantum computation require codes with an exceedingly rare property, a transverse non-Clifford gate. Codes with the desired property are presented for $d$-level, qudit, systems with prime $d$. The codes use $n=d-1$ qudits and can detect upto $\sim d/3$ errors. We quantify the performance of these codes for one approach to quantum computation, known as magic state distillation. Unlike prior work, we find performance is always enhanced by increasing $d$.

Fault-tolerant magic state preparation with flag qubits

Magic state distillation is one of the leading candidates for implementing universal fault-tolerant logical gates. However, the distillation circuits themselves are not fault-tolerant, so there is additional cost to first implement encoded Clifford gates with negligible error. In this paper we present a scheme to fault-tolerantly and directly prepare magic states using flag qubits. One of these schemes uses a single extra ancilla, even with noisy Clifford gates. We compare the physical qubit and gate cost of this scheme to the magic state distillation protocol of Meier, Eastin, and Knill, whic…

SpiderCSS：可扩展的容错CSS态制备

SpiderCSS: Scalable Fault-Tolerant CSS State Preparation

本文提出SpiderCSS，一种基于ZX-演算和故障等价重写的可扩展编译流水线，用于生成任意CSS码的容错态制备电路，实现CNOT最小化并显著降低深度和量子比特占用，同时平均逻辑错误率降低33.7%、接受率提高9.5%。

Single-shot fault-tolerant quantum error correction

Conventional quantum error correcting codes require multiple rounds of measurements to detect errors with enough confidence in fault-tolerant scenarios. Here I show that for suitable topological codes a single round of local measurements is enough. This feature is generic and is related to self-correction and confinement phenomena in the corresponding quantum Hamiltonian model. 3D gauge color codes exhibit this single-shot feature, which applies also to initialization and gauge-fixing. Assuming the time for efficient classical computations negligible, this yields a topological fault-tolerant q…