State preparation and measurement (SPAM) errors limit the performance of near-term quantum computers and their potential for practical application. SPAM errors are partly correctable after a calibration step that requires, for a complete implementation on a register of $n$ qubits, $2^n$ additional measurements. Here we introduce an approximate but efficient method for multiqubit SPAM error characterization and mitigation requiring the classical processing of $2^n \! \times 2^n$ matrices, but only $O(4^k n^2)$ measurements, where $k=O(1)$ is the number of qubits in a correlation volume. We demonstrate and validate the technique using an IBM Q processor on registers of 4 and 8 superconducting qubits.
Recently proposed implementations of quantum computer suffer from unavoidable interaction between quantum bits depending upon data being written in them. Novel procedure of avoiding multiqubit errors arising due to uncontrollable qubit-qubit interaction by using addititonal intermediate qubits is proposed. It is shown that the scheme requires only polynomial increase in number of qubits and algorithmic steps.
We review a time-dependent version of the Schrieffer-Wolff transformation that accounts for real-time control of system parameters, soon to be rendered possible on a broad basis due to technical progress. The dispersive regime of $N$ multilevel systems coupled to a cavity via a Jaynes-Cummings interaction is extended to the most general case. As a concrete example we rigorously apply the technique to dispersive two-qubit gates in a superconducting architecture, showing that fidelities based on previous models are off by up to $10^{-2}$, which is certainly relevant for high-fidelity gates compa…
Multiqubit Rydberg Gates for Quantum Error Correction
Peter Grünberg Institute, Theoretical Nanoelectronics, Forschungszentrum Jülich（于利希研究中心彼得·格林伯格研究所理论纳米电子学）；Institute for Quantum Information, RWTH Aachen University（亚琛工业大学量子信息研究所）；Institute for Theoretical Physics III and Center for Integrated Quantum Science and Technology, University of Stuttgart（斯图加特大学理论物理三系与集成量子科学与技术中心）
Quantum holonomic gates hold built-in resilience to local noises and provide a promising approach for implementing fault-tolerant quantum computation. We propose to realize high-fidelity holonomic $(N+1)$-qubit controlled gates using Rydberg atoms confined in optical arrays or superconducting circuits. We identify the scheme, deduce the effective multi-body Hamiltonian, and determine the working condition of the multiqubit gate. Uniquely, the multiqubit gate is immune to systematic errors, i.e., laser parameter fluctuations and motional dephasing, as the $N$ control atoms largely remain in the…
We propose a theoretical protocol to implement multiqubit geometric gates (i.e., the Mølmer-Sørensen gate) using photonic cat-state qubits. These cat-state qubits stored in high-$Q$ resonators are promising for hardware-efficient universal quantum computing. Specifically, in the limit of strong two-photon drivings, phase-flip errors of the cat-state qubits are effectively suppressed, leaving only a bit-flip error to be corrected. Because this dominant error commutes with the evolution operator, our protocol preserves the error bias, and, thus, can lower the code-capacity threshold for error co…
Several techniques have been recently introduced to mitigate errors in near-term quantum computers without the overhead required by quantum error correcting codes. While most of the focus has been on gate errors, measurement errors are significantly larger than gate errors on some platforms. A widely used {\it transition matrix error mitigation} (TMEM) technique uses measured transition probabilities between initial and final classical states to correct subsequently measured data. However from a rigorous perspective, the noisy measurement should be calibrated with perfectly prepared initial st…
The ability to perform gates in multiqubit systems that are robust to noise is of crucial importance for the advancement of quantum information technologies. However, finding control pulses that cancel noise while performing a gate is made difficult by the intractability of the time-dependent Schrodinger equation, especially in multiqubit systems. Here, we show that this issue can be sidestepped by using a formalism in which the cumulative error during a gate is represented geometrically as a curve in a multi-dimensional Euclidean space. Cancellation of noise errors to leading order correspond…
We examine the detailed scenario for implementing n-control-qubit Toffoli gates and select gates on ion-trap quantum computers, especially those that shuttle ions into interaction zones. We determine expected performance of these gates with realistic parameters for an ion-trap quantum computer and taking into account the time variation of the exchange integrals. This allows us to estimate the errors due to spin-phonon entanglement as well. While there are challenges with implementing these gates, because their performance always has some degree of error, they should be feasible on current hard…
We propose a scheme to generate Greenberger-Horne-Zeilinger (GHZ) state for N superconducting qubits in a circuit QED system. By sinusoidally modulating the qubit-qubit coupling, a synthetic magnetic field has been made which broken the time-reversal symmetry of the system. Directional rotation of qubit excitation can be realized in a three-qubit loop under the artificial magnetic field. Based on the special quality that the rotation of qubit excitation has different direction for single- and double-excitation loops, we can generate three-qubit GHZ state and extend this preparation method to a…
Due to its potential for implementing a scalable quantum computer, multiqubit Toffoli gate lies in the heart of quantum information processing. In this article, we demonstrate a multiqubit blockade gate with atoms arranged in a three-dimension spheroidal array. The gate performance is greatly improved by the method of optimizing control-qubit distributions on the spherical surface via evolutionary algorithm, which leads to an enhanced asymmetric Rydberg blockade. This spheroidal configuration, not only arises a well preservation for the dipole blockade energy between arbitrary control-target p…
We show how to perform error correction of single qubit dephasing by encoding a single qubit into a minimum of three. This may be performed in a manner closely analogous to classical error correction schemes. Further, the resulting quantum error correction schemes are trivially generalized to the minimal encoding of arbitrarily many qubits so as to allow for multiqubit dephasing correction under the sole condition that the environment acts independently on each qubit.
We introduce new measures of decoherence appropriate for evaluation of quantum computing designs. Environment-induced deviation of a quantum system's evolution from controlled dynamics is quantified by a single numerical measure. This measure is defined as a maximal norm of the density matrix deviation. We establish the property of additivity: in the regime of the onset of decoherence, the sum of the individual qubit error measures provides an estimate of the error for a several-qubit system. This property is illustrated by exact calculations for a spin-boson model.
General solutions to the quantum Rabi model involve subspaces with unbounded number of photons. However, for the multiqubit multimode case, we find special solutions with at most one photon for arbitrary number of qubits and photon modes. Unlike the Juddian solution, ours exists for arbitrary single qubit-photon coupling strength with constant eigenenergy. This corresponds to a horizontal line in the spectrum, while still being a qubit-photon entangled state. As a possible application, we propose an adiabatic scheme for the fast generation of arbitrary single-photon multimode W states with non…
As quantum circuits increase in size, it is critical to establish scalable multiqubit fidelity metrics. Here we investigate three-qubit randomized benchmarking (RB) with fixed-frequency transmon qubits coupled to a common bus with pairwise microwave-activated interactions (cross-resonance). We measure, for the first time, a three-qubit error per Clifford of 0.106 for all-to-all gate connectivity and 0.207 for linear gate connectivity. Furthermore, by introducing mixed dimensionality simultaneous RB --- simultaneous one- and two-qubit RB --- we show that the three-qubit errors can be predicted …
Magic states are eigenstates of non-Pauli operators. One way of suppressing errors present in magic states is to perform parity measurements in their non-Pauli eigenbasis and postselect on even parity. Here we develop new protocols based on non-Pauli parity checking, where the measurements are implemented with the aid of pre-distilled multiqubit resource states. This leads to a two step process: pre-distillation of multiqubit resource states, followed by implementation of the parity check. These protocols can prepare single-qubit magic states that enable direct injection of single-qubit axial …
Superconducting qubits, while promising for scalability and long coherence times, contain more than two energy levels, and therefore are susceptible to errors generated by the leakage of population outside of the computational subspace. Such leakage errors constitute a prominent roadblock towards fault-tolerant quantum computing (FTQC) with superconducting qubits. FTQC using topological codes is based on sequential measurements of multiqubit stabilizer operators. Here, we first propose a leakage-resilient procedure to perform repetitive measurements of multiqubit stabilizer operators, and then…
Peter Grünberg Institute, Forschungszentrum Jülich（彼得·格林堡研究所，吕贝克研究中心）；Jülich-Aachen Research Alliance (JARA)（吕贝克-亚琛研究联盟（JARA））；Fundamentals of Future Information Technologies（未来信息科技基础）；Institute for Quantum Information, RWTH Aachen University（量子信息研究所，亚琛RWTH大学）；Department of Software Technology, Delft University of Technology（软件技术系，代尔夫特理工大学）
The central challenge in building a quantum computer is error correction. Unlike classical bits, which are susceptible to only one type of error, quantum bits ("qubits") are susceptible to two types of error, corresponding to flips of the qubit state about the $X$- and $Z$-directions. While the Heisenberg Uncertainty Principle precludes simultaneous monitoring of $X$- and $Z$-flips on a single qubit, it is possible to encode quantum information in large arrays of entangled qubits that enable accurate monitoring of all errors in the system, provided the error rate is low. Another crucial requir…
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…
Crosstalk is a leading source of failure in multiqubit quantum information processors. It can arise from a wide range of disparate physical phenomena, and can introduce subtle correlations in the errors experienced by a device. Several hardware characterization protocols are able to detect the presence of crosstalk, but few provide sufficient information to distinguish various crosstalk errors from one another. In this article we describe how gate set tomography, a protocol for detailed characterization of quantum operations, can be used to identify and characterize crosstalk errors in quantum…
Randomized Benchmarking with Synthetic Quantum Circuits
Quantum Performance Laboratory, Sandia National Laboratories（桑迪亚国家实验室）；Department of Physics, University of Idaho（爱达荷大学）；HRL Laboratories, LLC（HRL实验室有限责任公司）
We present an error analysis of various tomographic protocols based on the linear inversion for the reconstruction of an unknown two-qubit state. We solve the problem of finding a tomographic protocol which is the most robust against errors in terms of the lowest value (i.e., equal to 1) of a condition number, as required by the Gastinel-Kahan theorem. In contrast, standard tomographic protocols, including those based on mutually unbiased bases, are nonoptimal for determining all 16 elements of an unknown two-qubit density matrix. Our method is based on the measurements of the 16 generalized P…
Quantum error correction is important to quantum information processing, which allows us to reliably process information encoded in quantum error correction codes. Efficient quantum error correction benefits from the knowledge of error rates. We propose a protocol for monitoring error rates in real time without interrupting the quantum error correction. Any adaptation of the quantum error correction code or its implementation circuit is not required. The protocol can be directly applied to the most advanced quantum error correction techniques, e.g. surface code. A Gaussian processes algorithm …
In addition to magnetic field and electric charge noise adversely affecting spin qubit operations, performing single-qubit gates on one of multiple coupled singlet-triplet qubits presents a new challenge---crosstalk, which is inevitable (and must be minimized) in any multiqubit quantum computing architecture. We develop a set of dynamically-corrected pulse sequences that are designed to cancel the effects of both types of noise (i.e., field and charge) as well as crosstalk to leading order, and provide parameters for these corrected sequences for all 24 of the single-qubit Clifford gates. We t…
Measurements that occur within the internal layers of a quantum circuit -- mid-circuit measurements -- are an important quantum computing primitive, most notably for quantum error correction. Mid-circuit measurements have both classical and quantum outputs, so they can be subject to error modes that do not exist for measurements that terminate quantum circuits. Here we show how to characterize mid-circuit measurements, modelled by quantum instruments, using a technique that we call quantum instrument linear gate set tomography (QILGST). We then apply this technique to characterize a dispersive…
The complexity of the error correction circuitry forces us to design quantum error correction codes capable of correcting a single error per error correction cycle. Yet, time-correlated error are common for physical implementations of quantum systems; an error corrected during the previous cycle may reoccur later due to physical processes specific for each physical implementation of the qubits. In this paper we study quantum error correction for a restricted class of time-correlated errors in a spin-boson model. The algorithm we propose allows the correction of two errors per error correction …
Operator quantum error correction provides a unified framework for the known techniques of quantum error correction such as the standard error correction model, the method of decoherence-free subspaces, and the noiseless subsystem method. We first show an example of a new quantum error correction scheme which can not be described by operation quantum error correction. Then we introduce a different notion of noiseless subsystems according to the example. Base on this notion, we present a more unified approach which incorporates operator quantum error correction as a special case. Moreover, we a…
Operator quantum error-correction is a technique for robustly storing quantum information in the presence of noise. It generalizes the standard theory of quantum error-correction, and provides a unified framework for topics such as quantum error-correction, decoherence-free subspaces, and noiseless subsystems. This paper develops (a) easily applied algebraic and information-theoretic conditions which characterize when operator quantum error-correction is feasible; (b) a representation theorem for a class of noise processes which can be corrected using operator quantum error-correction; and (c)…
We experimentally demonstrate a parametric iSWAP gate in a superconducting circuit based on a tunable coupler for achieving a continuous tunability to eliminate unwanted qubit interactions. We implement the twoqubit iSWAP gate by applying a fast-flux bias modulation pulse on the coupler to turn on parametric exchange interaction between computational qubits. The controllable interaction can provide an extra degree of freedom to verify the optimal condition for constructing the parametric gate. Aiming to fully investigate error sources of the two-qubit gates, we perform quantum process tomograp…
Fault tolerant protocol assumes the application of error correction after every quantum gate. However, correcting errors is costly in terms of time and number of qubits. Here we demonstrate that quantum error correction can be applied significantly less often with only a minimal loss of fidelity. This is done by simulating the implementation of 50 encoded, single-qubit, quantum gates within the [[7,1,3]] quantum error correction code in a noisy, non-equiprobable Pauli error environment with error correction being applied at different intervals. We find that applying error correction after ever…
Motivated by limitations and capabilities of neutral atom qubits, we examine whether measurement-free error correction can produce practical error thresholds. We show that this can be achieved by extracting redundant syndrome information, giving our procedure extra fault tolerance and eliminating the need for ancilla verification. The procedure is particularly favorable when multi-qubit gates are available for the correction step. Simulations of the bit-flip, Bacon-Shor, and Steane codes indicate that coherent error correction can produce threshold error rates that are on the order of $10^{-3}…
In this paper we show how the fault--tolerant error correction scheme recently proposed by DiVincenzo and Shor may be improved. Our scheme, unlike the earlier one, can also deal with a single error that might occur {\em during} the gate operations that are required for the implementation of the error correction and not only in--between the gates and hence presents an improvement towards enabling error correction and with it the practical possibility of some more involved quantum computations, such as e.g. factorization of large numbers.
I give an overview of the basic concepts behind quantum error correction and quantum fault tolerance. This includes the quantum error correction conditions, stabilizer codes, CSS codes, transversal gates, fault-tolerant error correction, and the threshold theorem.
Quantum error-correction routines are developed for continuous quantum variables such as position and momentum. The result of such analog quantum error correction is the construction of composite continuous quantum variables that are largely immune to the effects of noise and decoherence.
We propose a quantum error correction without error detection. A quantum state $ρ_0$ combined with an ancilla state $σ$ is encoded unitarily and an error operator is applied on the encoded state. The recovery operation then produces a tensor product state $ρ_0 \otimes σ'$. The decoding operation is combined with the recovery operation and the state $ρ_0$ is directly reproduced without referring to the code word. A higher rank projection operator required for a conventional operator quantum error correction is not necessary to implement. Encoding and the recovery operations are implemented with…
By taking into account the physical nature of quantum errors it is possible to improve the efficiency of quantum error correction. Here we consider an optimisation to conventional quantum error correction which involves exploiting asymmetries in the rates of X and Z errors by reducing the rate of X correction. As an example, we apply this optimisation to the [[7,1,3]] code and make a comparison with conventional quantum error correction. After two levels of concatenated error correction we demonstrate a circuit depth reduction of at least 43% and reduction in failure rate of at least 67%. This…
The standard quantum error correction protocols use projective measurements to extract the error syndromes from the encoded states. We consider the more general scenario of weak measurements, where only partial information about the error syndrome can be extracted from the encoded state. We construct a feedback protocol that probabilistically corrects the error based on the extracted information. Using numerical simulations of one-qubit error correction codes, we show that our error correction succeeds for a range of the weak measurement strength, where (a) the error rate is below the threshol…
It is argued that the existing schemes of fault-tolerant quantum computation designed for discrete-time models and based on quantum error correction fail for continuous-time Hamiltonian models even with Markovian noise.
From the set of operators for errors and its correction code, we introduce the so-called complete unitary transformation. It can be used for encoding while the inverse of it can be applied for correcting the errors of the encoded qubit. We show that this unitary protocol can be applied for any code which satisfies the quantum error correction condition.
This note presents a few observations on the nonlocal nature of quantum errors and the expected performance of the recently proposed quantum error-correction codes that are based on the assumption that the errors are either bit-flip or phase-flip or both.
Correcting errors is a vital but expensive component of fault tolerant quantum computation. Standard fault tolerant protocol assumes the implementation of error correction, via syndrome measurements and possible recovery operations, after every quantum gate. In fact, this is not necessary. Here we demonstrate that error correction should be applied more sparingly. We simulate encoded single-qubit rotations within the [[7,1,3]] code and show via fidelity measures that applying error correction after every gate is not desirable.
Inspired by Knill's scheme for message passing error detection, here we develop a scheme for message passing error correction for the nine-qubit Bacon-Shor code. We show that for two levels of concatenated error correction, where classical information obtained at the first level is used to help interpret the syndrome at the second level, our scheme will correct all cases with four physical errors. This results in a reduction of the logical failure rate relative to conventional error correction by a factor proportional to the reciprocal of the physical error rate.
Topological error correction provides an effective method to correct errors in quantum computation. It allows quantum computation to be implemented with higher error threshold and high tolerating loss rates. We present a topological a error correction scheme with continuous variables based on an eight-partite Gaussian cluster state. We show that topological quantum correlation between two modes can be protected against a single quadrature phase displacement error occurring on any mode and some of two errors occurring on two modes. More interestingly, some cases of errors occurring on three mod…
