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Quantum error correction offers a promising path for performing high fidelity quantum computations. Although fully fault-tolerant executions of algorithms remain unrealized, recent improvements in control electronics and quantum hardware enable increasingly advanced demonstrations of the necessary operations for error correction. Here, we perform quantum error correction on superconducting qubits connected in a heavy-hexagon lattice. We encode a logical qubit with distance three and perform several rounds of fault-tolerant syndrome measurements that allow for the correction of any single fault in the circuitry. Using real-time feedback, we reset syndrome and flag qubits conditionally after each syndrome extraction cycle. We report decoder dependent logical error, with average logical error per syndrome measurement in Z(X)-basis of ~0.040 (~0.088) and ~0.037 (~0.087) for matching and maximum likelihood decoders, respectively, on leakage post-selected data. Quantum error correction will be the key to allow large-scale quantum computing operations in the future. Here, the authors use a superconducting qubit system to demonstrate quantum error correction of a distance-three logical qubit in the heavy-hexagon subsystem code. [ABSTRACT FROM AUTHOR]

Quantum error correction (QEC) is a key technique for building scalable quantum computers that can be used to mitigate the effects of errors on physical quantum bits. Since quantum states are more or less affected by noise, errors are inevitable. Traditional QEC codes face huge challenges. Therefore, designing an error suppression algorithm based on neural networks (NN) and quantum topological error correction (QTEC) codes is particularly important for quantum teleportation. In this paper, QTEC codes: semion codes—a greater than 2 dimensional (2D) error correction code based on the double semion model—are used to suppress errors during quantum teleportation, using a NN to build a decoder based on semion codes and to simulate the quantum information error suppression process and the suppression effect. The proposed convolutional neural network (CNN) decoder is suitable for small distance topological semion codes. The aim is to optimize the NN for better decoder performance while deriving the relationship between decoder performance and slope and pseudothreshold during training and calculate the thresholds for different noise areas when the code distances are the same, P t h r e s h o l d = 0.082 for A r e a < 0.007 d B and P t h r e s h o l d = 0.096 for A r e a < 0.01 d B. This paper demonstrates the ability of CNNs to suppress errors in quantum transmission information and the great potential of NNs in the field of quantum computing. [ABSTRACT FROM AUTHOR]

Solving for quantum error correction remains one of the key challenges of quantum computing. Traditional decoding methods are limited by computing power and data scale, which restrict the decoding efficiency of color codes. There are many decoding methods that have been suggested to solve this problem. Machine learning is considered one of the most suitable solutions for decoding task of color code. We project the color code onto the surface code, use the deep Q network to iteratively train the decoding process of the color code and obtain the relationship between the inversion error rate and the logical error rate of the trained model and the performance of error correction. Our results show that through unsupervised learning, when iterative training is at least 300 times, a self-trained model can improve the error correction accuracy to 96.5%, and the error correction speed is about 13.8% higher than that of the traditional algorithm. We numerically show that our decoding method can achieve a fast prediction speed after training and a better error correction threshold. [ABSTRACT FROM AUTHOR]

Quantum computing can become scalable through error correction, but logical error rates only decrease with system size when physical errors are sufficiently uncorrelated. During computation, unused high energy levels of the qubits can become excited, creating leakage states that are long-lived and mobile. Particularly for superconducting transmon qubits, this leakage opens a path to errors that are correlated in space and time. Here, we report a reset protocol that returns a qubit to the ground state from all relevant higher level states. We test its performance with the bit-flip stabilizer code, a simplified version of the surface code for quantum error correction. We investigate the accumulation and dynamics of leakage during error correction. Using this protocol, we find lower rates of logical errors and an improved scaling and stability of error suppression with increasing qubit number. This demonstration provides a key step on the path towards scalable quantum computing. Correlated errors coming from leakage out of the computational subspace are an obstacle to fault-tolerant superconducting circuits. Here, the authors use a multi-level reset protocol to improve the performances of a bit-flip error correcting code by reducing the magnitude of correlations. [ABSTRACT FROM AUTHOR]

We propose a non-transversal but pieceable fault tolerant conversion circuit that is used to convert encoded information between five-qubit code and seven-qubit CSS code. Since a syndrome extraction circuit requiring fewer ancillary qubit resources would facilitate the realization of large-scale quantum computations, we further adapt a flag-assisted fault tolerant syndrome measurement scheme to reduce the cost of ancillary preparation. Numerical simulations are also performed to further analyze the performance of our conversion method. [ABSTRACT FROM AUTHOR]

The universal blind quantum computation (UBQC) is a scheme to allow a client to delegate a computation to a remote server while concealing the input, output and algorithm. However, the qubit errors are inevitable in the practical application. In this paper, a fault-tolerant quantum error correction code preparation protocol with weak coherent pulses is proposed for fault-tolerant UBQC. Furthermore, the ϵ -correctness and ϵ -blindness of the protocol are fully proven. The simulation results show that the required number of pulses in our protocol is much less than that of the remote blind qubit state preparation protocol with two decoy states in case of the same probability of successful preparation, and is closer to asymptotic case. [ABSTRACT FROM AUTHOR]

The concept of quantum computing has inspired a whole new generation of scientists, including physicists, engineers, and computer scientists, to fundamentally change the landscape of information technology. With experimental demonstrations stretching back more than two decades, the quantum computing community has achieved a major milestone over the past few years: the ability to build systems that are stretching the limits of what can be classically simulated, and which enable cloud-based research for a wide range of scientists, thus increasing the pool of talent exploring early quantum systems. While such noisy near-term quantum computing systems fall far short of the requirements for fault-tolerant systems, they provide unique test beds for exploring the opportunities for quantum applications. Here, we highlight an IBM-specific perspective of the facets associated with these systems, including quantum software, cloud access, benchmarking quantum systems, error correction and mitigation in such systems, understanding the complexity of quantum circuits, and how early quantum applications can run on near-term quantum computers. [ABSTRACT FROM AUTHOR]

We propose a family of non-locality unique games for two parties based on a square lattice on an arbitrary surface. We show that, due to structural similarities with error correction codes of Kitaev for fault tolerant quantum computation, the games have classical values computable in polynomial time for d = 2 measurement outcomes. By representing games in their graph form, for arbitrary d and underlying surface we provide their classification into equivalence classes with respect to relabeling of measurement outcomes, for a selected set of permutations which define the winning conditions. A case study of games with periodic boundary conditions is presented in order to verify their impact on classical and quantum values of the family of games. It suggests that quantum values suffer independently from presence of different winning conditions that can be imposed due to periodicity, as long as no local restrictions are in place. [ABSTRACT FROM AUTHOR]

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 particular, 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. [ABSTRACT FROM AUTHOR]

We show that the Mayers-Shor-Preskill approach and Renner’s approach to proving the security of quantum key distribution (QKD) are essentially the same. We begin our analysis by considering a special case of QKD called privacy amplification (PA). PA itself is an important building block of cryptography, both classical and quantum. The standard theoretical tool used for its security proof is called the leftover hashing lemma (LHL). We present a direct connection between the LHL and the coding theorem of a certain quantum error correction code. Then we apply this result to proving the equivalence between the two approaches to proving the security of QKD. [ABSTRACT FROM AUTHOR]

We study efficient quantum error correction schemes for the fully correlated channel on an n-qubit system with error operators that assume the form σ x ⊗ n , σ y ⊗ n , σ z ⊗ n . Previous schemes are improved to facilitate implementation. In particular, when n is odd and equals 2 k + 1 , we describe a quantum error correction scheme using one arbitrary qubit σ to protect the data state ρ in a 2k-qubit system. The encoding operation σ ⊗ ρ ↦ Φ (σ ⊗ ρ) only requires 3k CNOT gates (each with one control bit and one target bit). After the encoded state Φ (σ ⊗ ρ) goes through the channel, we can apply the inverse operation Φ - 1 to produce σ ~ ⊗ ρ so that a partial trace operation can recover ρ . When n is even and equals 2 k + 2 , we describe a hybrid quantum error correction scheme using any one of the two classical bits σ ∈ { | i j ⟩ ⟨ i j | : i , j ∈ { 0 , 1 } } to protect a 2k-qubit state ρ and two classical bits. The encoding operation σ ⊗ ρ ↦ Φ (σ ⊗ ρ) can be done by 3 k + 2 CNOT gates and a single-qubit Hadamard gate. After the encoded state Φ (σ ⊗ ρ) goes through the channel, we can apply the inverse operation Φ - 1 to produce σ ⊗ ρ so that a perfect protection of the two classical bits σ and the 2k-qubit state is achieved. If one uses an arbitrary two-qubit state σ , the same scheme will protect 2k-qubit states. The scheme was implemented using MATLAB, Mathematica, Python and the IBM's quantum computing framework qiskit. [ABSTRACT FROM AUTHOR]

Recently, there has been increasing interest in designing schemes for quantum computations that are robust against errors. Although considerable research has been devoted to quantum error correction schemes, much less attention has been paid to optimizing the speed it takes to perform a quantum computation and developing computation models that act on decoherence-free subspaces. Speeding up a quantum computation is important, because fewer errors are likely to result. Encoding quantum information in a decoherence-free subspace is also important, because errors would be inherently suppressed. In this paper, we consider quantum computation in a decoherence-free subspace and also optimize its speed. To achieve this, we perform certain single-qubit quantum computations by simply permuting the underlying qubits. In this paper, we make progress in understanding the extent in which quantum computation can be performed by permuting qubits. Namely, we provide two different subgroups of the single qubit Clifford group that can be computed solely by permutations. To make the quantum computation universal for one of the schemes, we rely on resonant couplings motivated from physics. Our first scheme potentially improves the speed in which a quantum computation can be done. We also reduce the problem of finding the existence of permutational Clifford gates to that finding an upper bound for the rank of certain matrices. [ABSTRACT FROM AUTHOR]

The accelerated progress in manufacturing noisy, intermediate-scale quantum (NISQ) computing hardware has opened the possibility of exploring its application in transforming approaches to solving computationally challenging problems. The important limitations common among all NISQ computing technologies are the absence of error correction and the short coherence time, which limit the computational power of these systems. Shortening the required time of a single run of a quantum algorithm is essential for reducing environment-induced errors and for the efficiency of the computation. We have investigated the ability of a variational version of adiabatic state preparation (ASP) to generate an accurate state more efficiently compared to existing adiabatic methods. The standard ASP method uses a time-dependent Hamiltonian, connecting the initial Hamiltonian with the final Hamiltonian. In the current approach, a navigator Hamiltonian is introduced which has a non-zero amplitude only in the middle of the annealing process. Both the initial and navigator Hamiltonians are determined using variational methods. A Hermitian cluster operator, inspired by coupled-cluster theory and truncated to single and double excitations/de-excitations, is used as a navigator Hamiltonian. A comparative study of our variational algorithm (VanQver) with that of standard ASP, starting with a Hartree–Fock Hamiltonian, is presented. The results indicate that the introduction of the navigator Hamiltonian significantly improves the annealing time required to achieve chemical accuracy by two to three orders of magnitude. The efficiency of the method is demonstrated in the ground-state energy estimation of molecular systems, namely, H2, P4, and LiH. [ABSTRACT FROM AUTHOR]

Instantaneous nonlocal quantum computation refers to a process in which spacelike separated parties simulate a nonlocal quantum operation on their joint systems through the consumption of pre-shared entanglement. To prevent a violation of causality, this simulation succeeds up to local errors that can only be corrected after the parties communicate classically with one another. However, this communication is non-interactive, and it involves just the broadcasting of local measurement outcomes. We refer to this operational paradigm as local operations and broadcast communication (LOBC) to distinguish it from the standard local operations and (interactive) classical communication (LOCC). In this paper, we show that an arbitrary two-qubit gate can be implemented by LOBC with $\epsilon $ -error using ${O}(\log (1/\epsilon))$ entangled bits (ebits). This offers an exponential improvement over the best known two-qubit protocols, whose ebit costs behave as ${O}(1/\epsilon)$. We also consider the family of binary controlled gates on dimensions ${d}_{A}\otimes {d}_{B}$. We find that any hermitian gate of this form can be implemented by LOBC using a single shared ebit. In sharp contrast, a lower bound of $\log {d}_{B}$ ebits is shown in the case of generic (i.e. non-hermitian) gates from this family, even when ${d}_{A}=2$. This demonstrates an unbounded gap between the entanglement costs of LOCC and LOBC gate implementation. Whereas previous lower bounds on the entanglement cost for instantaneous nonlocal computation restrict the minimum dimension of the needed entanglement, we bound its entanglement entropy. To our knowledge this is the first such lower bound of its kind. [ABSTRACT FROM AUTHOR]

The existence of quantum LDPC codes with minimal distance scaling linearly in the number of qubits is a central open problem in quantum information. Despite years of research good quantum LDPC codes are not known to exist, but at the very least it is known they cannot be defined on very regular topologies, like low-dimensional grids. In this work we establish a complementary result, showing that good quantum CSS codes which are sparsely generated require “structure” in the local terms that constrain the code space so as not to be “too-random” in a well-defined sense. To show this, we prove a weak converse to a theorem of Krasikov and Litsyn on weight distributions of classical codes due to which may be of independent interest: subspaces for which the distribution of weights in the dual space is approximately binomial have very few codewords of low weight, tantamount to having a non-negligible “approximate” minimal distance. While they may not have a large minimal non-zero weight, they still have very few words of low Hamming weight. [ABSTRACT FROM AUTHOR]

While single-shot detection of silicon spin qubits is now a laboratory routine, the need for quantum error correction in a large-scale quantum computing device demands a quantum non-demolition (QND) implementation. Unlike conventional counterparts, the QND spin readout imposes minimal disturbance to the probed spin polarization and can therefore be repeated to extinguish measurement errors. Here, we show that an electron spin qubit in silicon can be measured in a highly non-demolition manner by probing another electron spin in a neighboring dot Ising-coupled to the qubit spin. The high non-demolition fidelity (99% on average) enables over 20 readout repetitions of a single spin state, yielding an overall average measurement fidelity of up to 95% within 1.2 ms. We further demonstrate that our repetitive QND readout protocol can realize heralded high-fidelity (>99.6%) ground-state preparation. Our QND-based measurement and preparation, mediated by a second qubit of the same kind, will allow for a wide class of quantum information protocols with electron spins in silicon without compromising the architectural homogeneity. Conventional qubit readout methods in silicon spin qubits destroy the quantum state, precluding any further computations based on the outcome. Here, the authors demonstrate quantum non-demolition readout using a second qubit of the same kind, making for a scalable approach. [ABSTRACT FROM AUTHOR]

With rapid developments in quantum hardware comes a push towards the first practical applications. While fully fault-tolerant quantum computers are not yet realized, there may exist intermediate forms of error correction that enable practical applications. In this work, we consider the idea of post-processing error decoders using existing quantum codes, which mitigate errors on logical qubits using post-processing without explicit syndrome measurements or additional qubits beyond the encoding overhead. This greatly simplifies the experimental exploration of quantum codes on real, near-term devices, removing the need for locality of syndromes or fast feed-forward. We develop the theory of the method and demonstrate it on an example with the perfect [[5, 1, 3]] code, which exhibits a pseudo-threshold of p ≈ 0.50 under a single qubit depolarizing channel applied to all qubits. We also provide a demonstration of improved performance on an unencoded hydrogen molecule. Fault-tolerant quantum computation is still far, but there could be ways in which quantum error correction could improve currently available devices. Here, the authors show how to exploit existing quantum codes through only post-processing and random measurements in order to mitigate errors in NISQ devices. [ABSTRACT FROM AUTHOR]

We provide polylog sparse quantum codes for correcting the Erasure channel arbitrarily close to the capacity. Specifically, we provide the $[[n, k, d]]$ quantum stabilizer codes that correct for the erasure channel arbitrarily close to the capacity if the erasure probability is at least 0.33 and with a generating set $\langle S_{1}, S_{2}, \cdots {} S_{n-k} \rangle $ such that $|S_{i}|\leq {\mathrm{ log}} ^{2+\zeta }(n)$ for all $i$ and for any $\zeta > 0$ with high probability. In this paper, we show that the result of Delfosse et al. is tight: one can construct capacity approaching codes with weight almost $O(1)$. [ABSTRACT FROM AUTHOR]

Quantum computers can solve problems that are inefficiently solved by classical computers, such as integer factorization. A fully programmable quantum computer requires a quantum control microarchitecture that connects the quantum software and hardware. Previous research has proposed a Quantum Instruction Set Architecture (QISA) and a quantum control microarchitecture, which targets Noisy Intermediate-Scale Quantum (NISQ) devices without fault-tolerance. However, fault-tolerant (FT) quantum computing requires FT implementation of logical operations, and repeated quantum error correction, possibly at runtime. Though highly patterned, the amount of required (physical) operations to perform logical operations is ample, which cannot be well executed by existing quantum control microarchitectures. In this paper, we propose a control microarchitecture that can efficiently support fault-tolerant quantum computing based on the rotated planar surface code with logical operations implemented by lattice surgery. It highlights a two-level address mechanism which enables a clean compilation model for a large number of qubits, and microarchitectural support for quantum error correction at runtime, which can significantly reduce the quantum program codesize and present better scalability. [ABSTRACT FROM AUTHOR]

Quantum error correction protocols will play a central role in the realisation of quantum computing; the choice of error correction code will influence the full quantum computing stack, from the layout of qubits at the physical level to gate compilation strategies at the software level. As such, familiarity with quantum coding is an essential prerequisite for the understanding of current and future quantum computing architectures. In this review, we provide an introductory guide to the theory and implementation of quantum error correction codes. Where possible, fundamental concepts are described using the simplest examples of detection and correction codes, the working of which can be verified by hand. We outline the construction and operation of the surface code, the most widely pursued error correction protocol for experiment. Finally, we discuss issues that arise in the practical implementation of the surface code and other quantum error correction codes. [ABSTRACT FROM AUTHOR]

The reliable resource estimation and benchmarking of quantum algorithms is a critical component of the development cycle of viable quantum applications for quantum computers of all sizes. Determining resource bottlenecks in algorithms, especially when resource intensive error correction protocols are required, is crucial to reduce the cost of implementing viable algorithms on actual quantum hardware. [ABSTRACT FROM AUTHOR]

Quantum computers capable of solving classically intractable problems are under construction, and intermediate-scale devices are approaching completion. Current efforts to design large-scale devices require allocating immense resources to error correction, with the majority dedicated to the production of high-fidelity ancillary states known as magic-states. Leading techniques focus on dedicating a large, contiguous region of the processor as a single "magic-state distillation factory" responsible for meeting the magic-state demands of applications. In this work we design and analyze a set of optimized factory architectural layouts that divide a single factory into spatially distributed factories located throughout the processor. We find that distributed factory architectures minimize the space-time volume overhead imposed by distillation. Additionally, we find that the number of distributed components in each optimal configuration is sensitive to application characteristics and underlying physical device error rates. More specifically, we find that the rate at which T-gates are demanded by an application has a significant impact on the optimal distillation architecture. We develop an optimization procedure that discovers the optimal number of factory distillation rounds and number of output magic states per factory, as well as an overall system architecture that interacts with the factories. This yields between a 10x and 20x resource reduction compared to commonly accepted single factory designs. Performance is analyzed across representative application classes such as quantum simulation and quantum chemistry. [ABSTRACT FROM AUTHOR]

We give a scheme for interpreting shaded tangles as quantum circuits, with the property that if two shaded tangles are ambient isotopic, their corresponding computational effects are identical. We analyse 11 known quantum procedures in this way—including entanglement manipulation, error correction and teleportation—and in each case present a fully topological formal verification, yielding generalized procedures in some cases. We also use our methods to identify two new procedures, for topological state transfer and quantum error correction. Our formalism yields in some cases significant new insight into how the procedures work, including a description of quantum entanglement arising from topological entanglement of strands, and a description of quantum error correction where errors are ‘trapped by bubbles’ and removed from the shaded tangle. [ABSTRACT FROM AUTHOR]

We describe a computationally efficient heuristic algorithm based on a renormalization-group procedure which aims at solving the problem of finding a minimal surface given its boundary (curve) in any hypercubic lattice of dimension $D>2$. We use this algorithm to correct errors occurring in a four-dimensional variant of the toric code, having open as opposed to periodic boundaries. For a phenomenological error model which includes measurement errors we use a five-dimensional version of our algorithm, achieving a threshold of 4.35 ± 0.1%. For this error model, this is the highest known threshold of any topological code. Without measurement errors, a four-dimensional version of our algorithm can be used and we find a threshold of 7.3 ± 0.1%. For the gate-based depolarizing error model, we find a threshold of 0.31 ± 0.01% which is below the threshold found for the two-dimensional toric code. [ABSTRACT FROM AUTHOR]

A code-based cryptosystem can resist quantum-computing attacks. However, an original system based on the Goppa code has a large key size, which makes it unpractical in embedded devices with limited sources. Many special error-correcting codes have recently been developed to reduce the key size, and yet these systems are easily broken through side channel attacks, particularly differential power analysis (DPA) attacks, when they are applied to hardware devices. To address this problem, a higher-order masking scheme for a McEliece cryptosystem based on the quasi-dyadic moderate density parity check (QD-MDPC) code has been proposed. The proposed scheme has a small key size and is able to resist DPA attacks. In this paper, a novel McEliece cryptosystem based on the QD-MDPC code is demonstrated. The key size of this novel cryptosystem is reduced by 78 times, which meets the requirements of embedded devices. Further, based on the novel cryptosystem, a higher-order masking scheme was developed by constructing an extension Ishai-Sahai-Wagne (ISW) masking scheme. The authenticity and integrity analysis verify that the proposed scheme has higher security than conventional approaches. Finally, a side channel attack experiment was also conducted to verify that the novel masking system is able to defend against high-order DPA attacks on hardware devices. Based on the experimental validation, it can be concluded that the proposed higher-order masking scheme can be applied as an advanced protection solution for devices with limited resources. [ABSTRACT FROM AUTHOR]

Quantum computing has raised great interests for its potential to achieve an asymptotic speedup on specific problems. Current quantum devices suffer from noise which needs robust and scalable error-correcting schemes. Topological quantum error correction (TQEC) is among the most promising error-correcting techniques with exponential suppression of error with linear increase of space-time complexity. In this paper, we present the first work to explore space-time optimization between 1-D and 2-D architectures for TQEC circuits. We prove the NP-hardness of the qubit routing problem in the layout synthesis and propose an efficient algorithm to optimize space-time volumes for both 1-D and 2-D qubit architectures with promising experimental results. [ABSTRACT FROM AUTHOR]

In this paper, we improve the layered implementation of arbitrary stabilizer circuits introduced by Aaronson and Gottesman in Phys. Rev. A 70 (052328), 2004: to implement a general stabilizer circuit, we reduce their 11-stage computation -H-C-P-C-P-C-H-P-C-P-C- over the gate set consisting of Hadamard, controlled-NOT, and phase gates, into a 7-stage computation of the form -C-CZ-P-H-P-CZ-C-. We show arguments in support of using -CZ- stages over the -C- stages: not only the use of -CZ- stages allows a shorter layered expression, but -CZ- stages are simpler and appear to be easier to implement compared to the -C- stages. Based on this decomposition, we develop a two-qubit gate depth- $(14n{-}4)$ implementation of stabilizer circuits over the gate library $\{ \mathrm{H}, \mathrm{P}, \mathrm{CNOT}\}$ , executable in the Linear Nearest Neighbor (LNN) architecture, improving best previously known depth- $25n$ circuit, also executable in the LNN architecture. Our constructions rely on Bruhat decomposition of the symplectic group and on folding arbitrarily long sequences of the form (-P-C-) $^{m}$ into a three-stage computation -P-CZ-C-. Our results include the reduction of the 11-stage decomposition -H-C-P-C-P-C-H-P-C-P-C- into a 9-stage decomposition of the form -C-P-C-P-H-C-P-C-P-. This reduction is based on the Bruhat decomposition of the symplectic group. This result also implies a new normal form for stabilizer circuits. We show that a circuit in this normal form is optimal in the number of Hadamard gates used. We also show that the normal form has an asymptotically optimal number of parameters. [ABSTRACT FROM AUTHOR]

Topological error correction codes are promising candidates to protect quantum computations from the deteriorating effects of noise. While some codes provide high noise thresholds suitable for robust quantum memories, others allow straightforward gate implementation needed for data processing. To exploit the particular advantages of different topological codes for fault-tolerant quantum computation, it is necessary to be able to switch between them. Here we propose a practical solution, subsystem lattice surgery, which requires only two-body nearest-neighbor interactions in a fixed layout in addition to the indispensable error correction. This method can be used for the fault-tolerant transfer of quantum information between arbitrary topological subsystem codes in two dimensions and beyond. In particular, it can be employed to create a simple interface, a quantum bus, between noise resilient surface code memories and flexible color code processors. [ABSTRACT FROM AUTHOR]

Quantum information systems are expected to exhibit superiority compared with their classical counterparts. This superiority arises from the quantum coherences present in these quantum systems, which are obviously absent in classical ones. To exploit such quantum coherences, it is essential to control the phase information in the quantum state. The phase is analogue in nature, rather than binary. This makes quantum information technology fundamentally different from our classical digital information technology. In this paper, we analyse error sources and illustrate how these errors must be managed for the system to achieve the required fidelity and a quantum superiority. [ABSTRACT FROM AUTHOR]

We describe a general method for turning quantum circuits into sparse quantum subsystem codes. The idea is to turn each circuit element into a set of low-weight gauge generators that enforce the input–output relations of that circuit element. Using this prescription, we can map an arbitrary stabilizer code into a new subsystem code with the same distance and number of encoded qubits but where all the generators have constant weight, at the cost of adding some ancilla qubits. With an additional overhead of ancilla qubits, the new code can also be made spatially local. Applying our construction to certain concatenated stabilizer codes yields families of subsystem codes with constant-weight generators and with minimum distance d = n^1- \epsilon , where \epsilon = O(1/\sqrt \log n) . For spatially local codes in D dimensions, we nearly saturate a bound due to Bravyi and Terhal and achieve d = n^{1- \epsilon -1/D} . Previously the best code distance achievable with constant-weight generators in any dimension, due to Freedman, Meyer, and Luo, was O(\sqrt {n\log n})$ for a stabilizer code. [ABSTRACT FROM PUBLISHER]

We investigate the possibility to restore transient nutations of electron spin centers embedded in the solid using specific composite pulse sequences developed previously for the application in nuclear magnetic resonance spectroscopy. We treat two types of systematic errors simultaneously: (i) rotation angle errors related to the spatial distribution of microwave field amplitude in the sample volume, and (ii) off-resonance errors related to the spectral distribution of Larmor precession frequencies of the electron spin centers. Our direct simulations of the transient signal in erbium- and chromium-doped CaWO $$_{4}$$ crystal samples with and without error corrections show that the application of the selected composite pulse sequences can substantially increase the lifetime of Rabi oscillations. Finally, we discuss the applicability limitations of the studied pulse sequences for the use in solid-state electron paramagnetic resonance spectroscopy. [ABSTRACT FROM AUTHOR]

One central theme in quantum error-correction is to construct quantum codes that have a relatively large minimum distance. In this paper, we first present a construction of classical linear codes based on certain classes of polynomials. Through these classical linear codes, we are able to obtain some new quantum codes. It turns out that some of the quantum codes exhibited here have better parameters than the ones available in the literature. [ABSTRACT FROM PUBLISHER]

Cluster states can be exploited for some tasks such as topological one-way computation, quantum error correction, teleportation and dense coding. In this paper, we investigate and propose an arbitrated quantum signature scheme with cluster states. The cluster states are used for quantum key distribution and quantum signature. The proposed scheme can achieve an efficiency of 100 %. Finally, we also discuss its security against various attacks. [ABSTRACT FROM AUTHOR]

It is known that the security evaluation can be done by smoothing of Rényi entropy of order 2 in the classical and quantum settings when we apply universal \(_{2}\) hash functions. Using the smoothing of Rényi entropy of order 2, we derive security bounds for \(L_{1}\) distinguishability and modified mutual information criterion under the classical and quantum setting, and have derived these exponential decreasing rates. These results are extended to the case when we apply \(\varepsilon \) -almost dual universal \(_{2}\) hash functions. Furthermore, we apply this analysis to the secret key generation with error correction. [ABSTRACT FROM AUTHOR]

A long-standing open question about Gaussian continuous-variable cluster states is whether they enable fault-tolerant measurement-based quantum computation. The answer is yes. Initial squeezing in the cluster above a threshold value of 20.5 dB ensures that errors from finite squeezing acting on encoded qubits are below the fault-tolerance threshold of known qubit-based error-correcting codes. By concatenating with one of these codes and using ancilla-based error correction, fault-tolerant measurement-based quantum computation of theoretically indefinite length is possible with finitely squeezed cluster states. [ABSTRACT FROM AUTHOR]

A curve attaining the Hasse–Weil bound is called a maximal curve. Usually, classical error-correcting codes obtained from a maximal curve have good parameters. However, the quantum stabilizer codes obtained from such classical error-correcting codes via Euclidean or Hermitian self-orthogonality do not always possess good parameters. In this paper, the Hermitian self-orthogonality of algebraic geometry codes obtained from two maximal curves is investigated. It turns out that the stabilizer quantum codes produced from such Hermitian self-orthogonal classical codes have good parameters. [ABSTRACT FROM PUBLISHER]

Measurement-based quantum computation represents a powerful and flexible framework for quantum information processing, based on the notion of entangled quantum states as computational resources. The most prominent application is the one-way quantum computer, with the cluster state as its universal resource. Here we demonstrate the principles of measurement-based quantum computation using deterministically generated cluster states, in a system of trapped calcium ions. First we implement a universal set of operations for quantum computing. Second we demonstrate a family of measurement-based quantum error correction codes and show their improved performance as the code length is increased. The methods presented can be directly scaled up to generate graph states of several tens of qubits. [ABSTRACT FROM AUTHOR]