Error symmetrization in quantum computers

几何量子机器学习中的傅里叶对称化

Fourier Symmetrization for Geometric Quantum Machine Learning

本研究通过傅里叶表示分析对称性对量子傅里叶模型表达能力的影响，提出对称化与随机编码方法，并在量子物理信息神经网络中验证其有效性，降低误差。

Symmetrizing Cost of Quantum States

We introduce and analyze a task that we call symmetrization, in which a state of a quantum system, associated with a symmetry group, is transformed by a random unitary operation to a symmetric state. Each element of the unitary ensemble is required to be symmetry preserving, in the sense that it keeps the set of symmetric states invariant. We consider an asymptotic limit of infinitely many copies and vanishingly small error, and analyze the symmetrizing cost, that is, the minimum cost of randomness per copy required for symmetrization. We prove that the symmetrizing cost of an arbitrary quantu…

Symmetrizing Evolutions

We introduce quantum procedures for making $\cal G$-invariant the dynamics of an arbitrary quantum system S, where $\cal G$ is a finite group acting on the space state of S. Several applications of this idea are discussed. In particular when S is a N-qubit quantum computer interacting with its environment and $\cal G$ the symmetric group of qubit permutations, the resulting effective dynamics admits noiseless subspaces. Moreover it is shown that the recently introduced iterated-pulses schemes for reducing decoherence in quantum computers fit in this general framework. The noise-inducing compon…

Bang-Bang Operations from a Geometric Perspective

Strong, fast pulses, called ``bang-bang'' controls can be used to eliminate the effects of system-environment interactions. This method for preventing errors in quantum information processors is treated here in a geometric setting which leads to an intuitive perspective. Using this geometric description, we clarify the notion of group symmetrization as an averaging technique, and provide a geometric picture for evaluating errors due to imperfect bang-bang controls. This will provide additional support for the usefulness of such controls as a means for providing more reliable quantum informatio…

The Quantum Interference Computer: error correction

An error correcting mechanism is proposed in the context of the Quantum Interference Computer approach

Stabilization of Quantum Information: A Unified Dynamical-Algebraic Approach

The notion of symmetry is shown to be at the heart of all error correction/avoidance strategies for preserving quantum coherence of an open quantum system S e.g., a quantum computer. The existence of a non-trivial group of symmetries of the dynamical algebra of S provides state-space sectors immune to decoherence. Such noiseless sectors, that can be viewed as a noncommutative version of the pointer basis, are shown to support universal quantum computation and to be robust against perturbations. When the required symmetry is not present one can generate it artificially resorting to active symme…

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.

Demonstration of teleportation-based error correction in the IBM quantum computer

Quantum error correcting codes (QECC) are the key ingredients both for fault-tolerant quantum computation and quantum communication. Teleportation-based error correction (TEC) helps in detecting and correcting operational and erasure errors by performing X and Z measurements during teleportation. Here we demonstrate the TEC protocol for the detection and correction of a single bit-flip error by proposing a new quantum circuit. A single phase-flip error can also be detected and corrected using the above protocol. For the first time, we illustrate detection and correction of erasure error in the…

Analytic solutions for quantum logic gates and modeling pulse errors in a quantum computer with a Heisenberg interaction

We analyze analytically and numerically quantum logic gates in a one-dimensional spin chain with Heisenberg interaction. Analytic solutions for basic one-qubit gates and swap gate are obtained for a quantum computer based on logical qubits. We calculated the errors caused by imperfect pulses which implement the quantum logic gates. It is numerically demonstrated that the probability error is proportional to $ε^4$, while the phase error is proportional to $ε$, where $ε$ is the characteristic deviation from the perfect pulse duration.

Categorizing Readout Error Correlations on Near Term Quantum Computers

Readout errors are a significant source of noise for near term quantum computers. A variety of methods have been proposed to mitigate these errors using classical post processing. For a system with $n$ qubits, the entire readout error profile is specified by a $2^n\times 2^n$ matrix. Recent proposals to use sub-exponential approximations rely on small and/or short-ranged error correlations. In this paper, we introduce and demonstrate a methodology to categorize and quantify multiqubit readout error correlations. Two distinct types of error correlations are considered: sensitivity of the measur…

Analysis of Error Propagation in Quantum Computers

Most quantum gate errors can be characterized by two error models, namely the probabilistic error model and the Kraus error model. We proved that for a quantum circuit with either of those two models or a mix of both, the propagation error in terms of Frobenius norm is upper bounded by $2(1 - (1 - r)^m)$, where $0 \le r < 1$ is a constant independent of the qubit number and circuit depth, and $m$ is the number of gates in the circuit. Numerical experiments of synthetic quantum circuits and quantum Fourier transform circuits are performed on the simulator of the IBM Vigo quantum computer to ver…

Error Rate of the Kane Quantum Computer CNOT Gate in the Presence of Dephasing

We study the error rate of CNOT operations in the Kane solid state quantum computer architecture. A spin Hamiltonian is used to describe the system. Dephasing is included as exponential decay of the off diagonal elements of the system's density matrix. Using available spin echo decay data, the CNOT error rate is estimated at approsimately 10^{-3}.

Error-analysis for the Sorkin and Peres tests performed on a quantum computer

We use quantum computers to test the foundations of quantum mechanics through quantum algorithms that implement some of the experimental tests as the basis of the theory's postulates. These algorithms can be used as a test of the physical theory under the premise of a perfect hardware or as a test of the hardware under the premise that quantum theory is correct. In this paper, we show how the algorithms can be used to test the efficacy of a quantum computer in obeying the postulates of quantum mechanics. We study the effect of different types of errors on the results of experimental tests of t…

Error mitigation increases the effective quantum volume of quantum computers

Quantum volume is a single-number metric which, loosely speaking, reports the number of usable qubits on a quantum computer. While improvements to the underlying hardware are a direct means of increasing quantum volume, the metric is "full-stack" and has also been increased by improvements to software, notably compilers. We extend this latter direction by demonstrating that error mitigation, a type of indirect compilation, increases the effective quantum volume of several quantum computers. Importantly, this increase occurs while taking the same number of overall samples. We encourage the adop…

Manifold algorithmic errors in quantum computers with static internal imperfections

The inevitable existence of static internal imperfections and residual interactions in some quantum computer architectures result in internal decoherence, dissipation, and destructive unitary shifts of active algorithms. By exact numerical simulations we determine the relative importance and origin of these errors for a Josephson charge qubit quantum computer. In particular we determine that the dynamics of a CNOT gate interacting with its idle neighboring qubits via native residual coupling behaves much like a perturbed kicked top in the exponential decay regime, where fidelity decay is only …

Demonstration of a general fault-tolerant quantum error detection code for (2n+1)-qubit entangled state on IBM 16-qubit quantum computer

Quantum error detection has always been a fundamental challenge in a fault-tolerant quantum computer. Hence, it is of immense importance to detect and deal with arbitrary errors to efficiently perform quantum computation. Several error detection codes have been proposed and realized for lower number of qubit systems. Here we present an error detection code for a (2n+1)-qubit entangled state using two syndrome qubits and simulate it on IBM's 16-qubit quantum computer for a 13-qubit entangled system. The code is able to detect an arbitrary quantum error in any one of the first 2n qubits of the (…

Error correction in ensemble registers for quantum repeaters and quantum computers

We propose to use a collective excitation blockade mechanism to identify errors that occur due to disturbances of single atoms in ensemble quantum registers where qubits are stored in the collective population of different internal atomic states. A simple error correction procedure and a simple decoherence-free encoding of ensemble qubits in the hyperfine states of alkali atoms are presented.

An analytical error model for quantum computer simulation

Quantum computers (QCs) must implement quantum error correcting codes (QECCs) to protect their logical qubits from errors, and modeling the effectiveness of QECCs on QCs is an important problem for evaluating the QC architecture. The previously developed Monte Carlo (MC) error models may take days or weeks of execution to produce an accurate result due to their random sampling approach. We present an alternative analytical error model that generates, over the course of executing the quantum program, a probability tree of the QC's error states. By calculating the fidelity of the quantum program…

Automated Error Correction in IBM Quantum Computer and Explicit Generalization

Experimental realization of automated error correction is demonstrated through IBM Quantum Experience for Bell and GHZ states using a measurement based approach upon ancilla qubits. The measurement automatically activates error correcting unitary operations to restore the system to its original entangled state. We illustrate the algorithm for the maximally entangled qudit case by applying appropriate Hadamard and Controlled-NOT gates.

Finding the ground state of the Hubbard model by variational methods on a quantum computer with gate errors

A key goal of digital quantum computing is the simulation of fermionic systems such as molecules or the Hubbard model. Unfortunately, for present and near-future quantum computers the use of quantum error correction schemes is still out of reach. Hence, the finite error rate limits the use of quantum computers to algorithms with a low number of gates. The variational Hamiltonian ansatz (VHA) has been shown to produce the ground state in good approximation in a manageable number of steps. Here we study explicitly the effect of gate errors on its performance. The VHA is inspired by the adiabatic…

Introduction to error correcting codes in quantum computers

The goal of this paper is to review the theoretical basis for achieving a faithful quantum information transmission and processing in the presence of noise. Initially encoding and decoding, implementing gates and quantum error correction will be considered error free. Finally we will relax this non realistic assumption, introducing the quantum fault-tolerant concept. The existence of an error threshold permits to conclude that there is no physical law preventing a quantum computer from being built. An error model based on the depolarizing channel will be able to provide a simple estimation of …

Effective error-suppression scheme for reversible quantum computer

We construct a new error-suppression scheme that makes use of the adjoint of reversible quantum algorithms. For decoherence induced errors such as depolarization, it is presented that provided the depolarization error probability is less than 1, our scheme can exponentially reduce the final output error rate to zero using a number of cycles, and the output state can be coherently sent to another stage of quantum computation process. Besides, experimental set-ups via optical approach have been proposed using Grover's search algorithm as an example. Some further discussion on the benefits and li…

部分纠错量子计算机的错误缓解

Error mitigation for partially error-corrected quantum computers

针对部分纠错量子计算机，提出基于信道凸组合电路的错误缓解方法，利用逻辑辅助量子比特在时空权衡下将采样复杂度降至 $O(1)$，并证明其在开放量子系统模拟中能显著减少门数量。

量子计算机读出误差缓解的信息论方法

Information theoretic approach to readout error mitigation for quantum computers

本文从信息论分析推导出迭代贝叶斯展开的读出误差缓解方法，区分了结构性缓解与非结构性缓解，并利用127比特量子计算机的GHZ态制备数据验证了局部边缘分布缓解的有效性。

损耗和退相干对简单量子计算机的影响

Effects of Loss and Decoherence on a Simple Quantum Computer

我们研究了损耗（振幅阻尼）和退相干（相位阻尼）对一个简单的量子计算机的性能的影响，它解决了一位Deutsch问题。这台机器的组成部分是分束器和非线性光学克尔盒，但误差主要来自后者。我们开发的模型来描述这些错误对量子光学Fredkin门的影响。结果被用来分析在一个完整的量子计算机可能的纠错策略。我们发现，由于损失的错误可以完全避免适当的设计技术，而退相干可以部分处理使用投影误差校正。

Calculation of $π$ on the IBM quantum computer and the accuracy of one-qubit operations

A quantum algorithm for the calculation of $π$ is proposed and implemented on the five-qubit IBM quantum computer with superconducting qubits. We find $π=3.157\pm0.017$. The error is due to the noise of quantum one-qubit operations and measurements. The results can be used for estimating the errors of the quantum computer and suggest that the errors are purely random.

Scalable mitigation of measurement errors on quantum computers

We present a method for mitigating measurement errors on quantum computing platforms that does not form the full assignment matrix, or its inverse, and works in a subspace defined by the noisy input bit-strings. This method accommodates both uncorrelated and correlated errors, and allows for computing accurate error bounds. Additionally, we detail a matrix-free preconditioned iterative solution method that converges in $\mathcal{O}(1)$ steps that is performant and uses orders of magnitude less memory than direct factorization. We demonstrate the validity of our method, and mitigate errors in a…

Automated error correction in superdense coding, with implementation on superconducting quantum computer

Construction of a fault-tolerant quantum computer remains a challenging problem due to unavoidable noise in quantum states and the fragility of quantum entanglement. However, most of the error-correcting codes increases the complexity of the algorithms, thereby decreasing any quantum advantage. Here we present a task-specific error-correction technique that provides a complete protection over a restricted set of quantum states. Specifically, we give an automated error correction in Superdense Coding algorithms utilizing n-qubit generalized Bell states. At its core, it is based on non-destructi…

5-qubit quantum error correction in a charge qubit quantum computer

We implement the DiVincenzo-Shor 5 qubit quantum error correcting code into a solid-state quantum register. The quantum register is a multi charge-qubit system in a semiconductor environment, where the main sources of noise are phase decoherence and relaxation. We evaluate the decay of the density matrix for this multi-qubit system and perform regular quantum error corrections. The performance of the error correction in this realistic system is found to yield an improvement of the fidelity. The fidelity can be maintained arbitrarily close to one by sufficiently increasing the frequency of erro…

How Quantum Computers Can Fail

We propose and discuss two postulates on the nature of errors in highly correlated noisy physical stochastic systems. The first postulate asserts that errors for a pair of substantially correlated elements are themselves substantially correlated. The second postulate asserts that in a noisy system with many highly correlated elements there will be a strong effect of error synchronization. These postulates appear to be damaging for quantum computers.

通过量子纠错校验子测量探究量子计算机的误差结构

Quantum computer error structure probed by quantum error correction syndrome measurements

本研究通过在IBM超导量子器件上开展最大23比特的重六角码校验子测量实验，验证现有简化误差模型的有效性，发现噪声存在偏置、非均匀性与时间相关性，为定制化纠错码与解码器开发提供依据。

Simulating the Transverse Ising Model on a Quantum Computer: Error Correction with the Surface Code

We estimate the resource requirements for the quantum simulation of the ground state energy of the one dimensional quantum transverse Ising model (TIM), based on the surface code implementation of a fault tolerant quantum computer. The surface code approach has one of the highest known tolerable error rates (1%) which makes it currently one of the most practical quantum computing schemes. Compared to results of the same model using the concatenated Steane code, the current results indicate that the simulation time is comparable but the number of physical qubits for the surface code is 2-3 orde…

Reliable Quantum Computers

The new field of quantum error correction has developed spectacularly since its origin less than two years ago. Encoded quantum information can be protected from errors that arise due to uncontrolled interactions with the environment. Recovery from errors can work effectively even if occasional mistakes occur during the recovery procedure. Furthermore, encoded quantum information can be processed without serious propagation of errors. Hence, an arbitrarily long quantum computation can be performed reliably, provided that the average probability of error per quantum gate is less than a certain …

Comments on `Stable Quantum Computation of Unstable Classical Chaos,' `Efficient Quantum Computing Insensitive to Phase Errors,' and `Quantum Computer Inverting Time Arrow for Macroscopic Systems'

Recently, the paper B. Georgeot and D.L. Shepelyansky, Phys. Rev. Lett._86_, 5393 (2001) has been criticized [C. Zalka, quant-ph/0110019; L. Di'osi, quant-ph/0110026]. The Letter claims an exponential speedup and reduction in error sensitivity, when phase-space density evolution under the Arnold cat map is performed on a quantum computer. On the one hand, some points have not yet been made by previous respondents. On the other, the authors' reaction [quant-ph/0110142] raises new issues. The present note addresses both.

Strictly contractive quantum channels and physically realizable quantum computers

We study the robustness of quantum computers under the influence of errors modelled by strictly contractive channels. A channel $T$ is defined to be strictly contractive if, for any pair of density operators $ρ,σ$ in its domain, $\| Tρ- Tσ\|_1 \le k \| ρ-σ\|_1$ for some $0 \le k < 1$ (here $\| \cdot \|_1$ denotes the trace norm). In other words, strictly contractive channels render the states of the computer less distinguishable in the sense of quantum detection theory. Starting from the premise that all experimental procedures can be carried out with finite precision, we argue that there exis…

量子计算机上态制备误差与测量误差的高效分离量化及其缓解

Efficient separate quantification of state preparation errors and measurement errors on quantum computers and their mitigation

针对量子计算机中态制备误差与测量误差难以分离的问题，本文提出一种资源高效的分别量化方法，并设计线性复杂度的态制备误差缓解方案，使结果保真度较标准方案提升一个数量级，且对门噪声具有鲁棒性。

Qubit metrology for building a fault-tolerant quantum computer

Recent progress in quantum information has led to the start of several large national and industrial efforts to build a quantum computer. Researchers are now working to overcome many scientific and technological challenges. The program's biggest obstacle, a potential showstopper for the entire effort, is the need for high-fidelity qubit operations in a scalable architecture. This challenge arises from the fundamental fragility of quantum information, which can only be overcome with quantum error correction. In a fault-tolerant quantum computer the qubits and their logic interactions must have …

Scalable Error Correction in Distributed Ion Trap Computers

A major challenge for quantum computation in ion trap systems is scalable integration of error correction and fault tolerance. We analyze a distributed architecture with rapid high fidelity local control within nodes and entangled links between nodes alleviating long-distance transport. We demonstrate fault-tolerant operator measurements which are used for error correction and non-local gates. This scheme is readily applied to linear ion traps which cannot be scaled up beyond a few ions per individual trap but which have access to a probabilistic entanglement mechanism. A proof-of-concept syst…

Numerical simulation of information recovery in quantum computers

Decoherence is the main problem to be solved before quantum computers can be built. To control decoherence, it is possible to use error correction methods, but these methods are themselves noisy quantum computation processes. In this work we study the ability of Steane's and Shor's fault-tolerant recovering methods, as well a modification of Steane's ancilla network, to correct errors in qubits. We test a way to measure correctly ancilla's fidelity for these methods, and state the possibility of carrying out an effective error correction through a noisy quantum channel, even using noisy error …

Quantum computers can search rapidly by using almost any selective transformations

The search problem is to find a state satisfying certain properties out of a given set. Grover's algorithm drives a quantum computer from a prepared initial state to the target state and solves the problem quadratically faster than a classical computer. The algorithm uses selective transformations to distinguish the initial state and target state from other states. It does not succeed unless the selective transformations are very close to phase-inversions. Here we show a way to go beyond this limitation. An important application lies in quantum error-correction, where the errors can cause the …

Error Mitigation in Quantum Computers subject to Spatially Correlated Noise

The most common error models for quantum computers assume the independence of errors on different qubits. However, most noise mechanisms have some correlations in space. We show how to improve quantum information processing for few-qubit systems when spatial correlations are present. This starts with strategies to measure the correlations. Once the correlations have been determined, we can give criteria to assess the suitability of candidate quantum circuits to carry out a given task. This is achieved by defining measures of decoherence that are local in Hilbert space, identifying "good" and "…

Simulation of static and random errors on Grover's search algorithm implemented in a Ising nuclear spin chain quantum computer with few qubits

We consider Grover's search algorithm on a model quantum computer implemented on a chain of four or five nuclear spins with first and second neighbour Ising interactions. Noise is introduced into the system in terms of random fluctuations of the external fields. By averaging over many repetitions of the algorithm, the output state becomes effectively a mixed state. We study its overlap with the nominal output state of the algorithm, which is called fidelity. We find either an exponential or a Gaussian decay for the fidelity as a function of the strength of the noise, depending on the type of n…

近期量子计算机上开放量子系统的长时间误差缓解模拟

Long-Time Error-Mitigating Simulation of Open Quantum Systems on Near Term Quantum Computers

本文在IBM量子硬件上模拟开放量子系统，展示深层电路下长时间误差缓解能力，证明耗散算法优于非耗散算法，为驱动-耗散量子多体问题的求解带来希望。

一种面向量子比特量子计算机的误差感知和自适应可观测量估计方法

An Error-aware and Adaptive Method for the Estimation of Quantum Observables on Qudit-Based Quantum Computers

本文提出AQUIRE方法，通过构建贝叶斯模型准确估计量子比特量子计算机上可观测量的均值和误差，并通过数值模拟和离子阱量子处理器实验验证其有效性。

Simulating the Effect of Decoherence and Inaccuracies on a Quantum Computer

A Quantum Computer is a new type of computer which can solve problems such as factoring and database search very efficiently. The usefulness of a quantum computer is limited by the effect of two different types of errors, decoherence and inaccuracies. In this paper we show the results of simulations of a quantum computer which consider both decoherence and inaccuracies. We simulate circuits which factor the numbers 15, 21, 35, and 57 as well as circuits which use database search to solve the circuit satisfaction problem. Our simulations show that the error rate per gate is on the order of 10^-…

Numerical Simulation of Decoherence-control Processes in Quantum Computers

Using a numerical simulation of the evolution of a qubit interacting with the environment we show that quantum error detection and correction can work effectively even when the recovery procedure introduces errors.

Creation of entanglement in a scalable spin quantum computer with long-range dipole-dipole interaction between qubits

Creation of entanglement is considered theoretically and numerically in an ensemble of spin chains with dipole-dipole interaction between the spins. The unwanted effect of the long-range dipole interaction is compensated by the optimal choice of the parameters of radio-frequency pulses implementing the protocol. The errors caused by (i) the influence of the environment,(ii) non-selective excitations, (iii) influence of different spin chains on each other, (iv) displacements of qubits from their perfect locations, and (v) fluctuations of the external magnetic field are estimated analytically an…

Simulation of Quantum Error Correcting Code

This study considers implementations of error correction in a simulation language on a classical computer. Error correction will be necessarily in quantum computing and quantum information. We will give some examples of the implementations of some error correction codes. These implementations will be made in a more general quantum simulation language on a classical computer in the language Mathematica. The intention of this research is to develop a programming language that is able to make simulations of all quantum algorithms and error corrections in the same framework. The program code imple…

Stability of quantum Fourier transformation on Ising quantum computer

We analyze the influence of errors on the implementation of the quantum Fourier transformation (QFT) on the Ising quantum computer (IQC). Two kinds of errors are studied: (i) due to spurious transitions caused by pulses and (ii) due to external perturbation. The scaling of errors with system parameters and number of qubits is explained. We use two different procedures to fight each of them. To suppress spurious transitions we use correcting pulses (generalized $2πk$ method) while to suppress errors due to external perturbation we use an improved QFT algorithm. As a result, the fidelity of quan…

Characterization of how errors accumulate in quantum computers

We study the achievements of quantum circuits comprised of several one- and two-qubit gates. Quantum process matrices are determined for the basic one- and two-qubit gate operations and concatenated to yield the process matrix of the combined quantum circuit. Examples are given of process matrices obtained by a Monte Carlo wave function analysis of Rydberg blockade gates in neutral atoms. Our analysis is ideally suited to compare different implementations of the same process. In particular, we show that the three-qubit Toffoli gate facilitated by the simultaneous interaction between all atoms …

噪声量子计算机上量子模拟的最优阶 Trotter-Suzuki 分解

Optimal-order Trotter-Suzuki decomposition for quantum simulation on noisy quantum computers

本文研究在噪声量子计算机上使用高阶 Trotter-Suzuki 分解模拟量子系统的条件，发现当门错误降低约一个数量级时，高阶 Trotterization 可最小化数学与物理总误差。

A Parallel Quantum Computer Simulator

A Quantum Computer is a new type of computer which can efficiently solve complex problems such as prime factorization. A quantum computer threatens the security of public key encryption systems because these systems rely on the fact that prime factorization is computationally difficult. Errors limit the effectiveness of quantum computers. Because of the exponential nature of quantum com puters, simulating the effect of errors on them requires a vast amount of processing and memory resources. In this paper we describe a parallel simulator which accesses the feasibility of quantum computers. We …

Models to Reduce the Complexity of Simulating a Quantum Computer

Recently Quantum Computation has generated a lot of interest due to the discovery of a quantum algorithm which can factor large numbers in polynomial time. The usefulness of a quantum com puter is limited by the effect of errors. Simulation is a useful tool for determining the feasibility of quantum computers in the presence of errors. The size of a quantum computer that can be simulat ed is small because faithfully modeling a quantum computer requires an exponential amount of storage and number of operations. In this paper we define simulation models to study the feasibility of quantum comput…

Characterizing large-scale quantum computers via cycle benchmarking

Quantum computers promise to solve certain problems more efficiently than their digital counterparts. A major challenge towards practically useful quantum computing is characterizing and reducing the various errors that accumulate during an algorithm running on large-scale processors. Current characterization techniques are unable to adequately account for the exponentially large set of potential errors, including cross-talk and other correlated noise sources. Here we develop cycle benchmarking, a rigorous and practically scalable protocol for characterizing local and global errors across mult…