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arXiv:2108.10927 · Physical Review A

Error mitigation for variational quantum algorithms through mid-circuit measurements

Ludmila Botelho, Adam Glos, Akash Kundu, Jarosław Adam Miszczak, Özlem Salehi, Zoltán Zimborás

Published Feb 28, 2022• 54 citations
arXivHTMLPDF

Abstract

Noisy intermediate-scale quantum algorithms require novel paradigms of error mitigation. To obtain noise-robust quantum computers, each logical qubit is equipped with hundreds or thousands of physical qubits. However, it is not possible to use memory-consuming techniques for current quantum devices having at most hundreds or thousands of physical qubits on their own. For specific problems, valid quantum states have a unique structure as in the case of Fock states and $W$ states where the Hamming weight is fixed, and the evolution takes place in a smaller subspace of the full Hilbert space. With this preknowledge, some errors can be detected during the evolution of the circuit, by filtering the states not obeying the pattern through postselection. In this paper, we present mid-circuit postselection schemes for frequently used encodings such as one-hot, binary, gray, and domain-wall encoding. For the particular subspace of one-hot states, we propose a method that works by compressing the full Hilbert space to a smaller subspace, allowing projecting to the desired subspace without using any ancilla qubits. We demonstrate the effectiveness of the approach for the quantum alternating operator ansatz algorithm. Our method is particularly suitable for the currently available hardware, where measuring and resetting are possible, but classical conditional operators are not.

DOI
10.1103/physreva.105.022441

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Noisy intermediate-scale quantum algorithms require novel paradigms of error mitigation. To obtain noise-robust quantum computers, each logical qubit is equipped with hundreds or thousands of physical qubits. However, it is not possible to use memory-consuming techniques for current quantum devices having at most hundreds or thousands of physical qubits on their own. For specific problems, valid quantum states have a unique structure as in the case of Fock states and WWW states where the Hamming weight is fixed, and the evolution takes place in a smaller subspace of the full Hilbert space. With this preknowledge, some errors can be detected during the evolution of the circuit, by filtering the states not obeying the pattern through postselection. In this paper, we present mid-circuit postselection schemes for frequently used encodings such as one-hot, binary, gray, and domain-wall encoding. For the particular subspace of one-hot states, we propose a method that works by compressing the full Hilbert space to a smaller subspace, allowing projecting to the desired subspace without using any ancilla qubits. We demonstrate the effectiveness of the approach for the quantum alternating operator ansatz algorithm. Our method is particularly suitable for the currently available hardware, where measuring and resetting are possible, but classical conditional operators are not.
Current quantum computing platforms suffer from readout errors, where faulty measurement outcomes are reported by the device. These errors are particularly harmful in quantum programs that rely on branch statements, where operations in later parts of the program are dynamically determined by mid-circuit measurements. We propose a general protocol for mitigating mid-circuit measurement errors. Our solution works for any number of mid-circuit measurement and feedforward layers without increasing circuit depth or two-qubit gate counts, is efficient under structural assumptions on error channels, and is suitable for immediate application on noisy intermediate-scale quantum devices. We experimentally demonstrate up to a ∼ 60 % reduction in error on superconducting quantum processors across several practically relevant feedforward circuits, including dynamic qubit resets, shallow-depth Greenberger–Horne–Zeilinger state preparation, and multistage quantum teleportation. This work paves the way for more error-resilient adaptive quantum circuits, crucial for current and future quantum computing applications.
Suppressing noise in physical systems is of fundamental importance.As quantum computers mature, quantum error correcting codes (QECs) will be adopted in order to suppress errors to any desired level.However in the noisy, intermediate-scale quantum (NISQ) era, the complexity and scale required to adopt even the smallest QEC is prohibitive: a single logical qubit needs to be encoded into many thousands of physical qubits.Here we show that, for the crucial case of estimating expectation values of observables (key to almost all NISQ algorithms) one can indeed achieve an effective exponential suppression.We take n independently prepared circuit outputs to create a state whose symmetries prevent errors from contributing bias to the expected value.The approach is very well suited for current and near-term quantum devices as it is modular in the main computation and requires only a shallow circuit that bridges the n copies immediately prior to measurement.Using no more than four circuit copies, we confirm error suppression below 10 -6 for circuits consisting of several hundred noisy gates (2-qubit gate error 0.5%) in numerical simulations validating our approach.
In this paper, we describe and experimentally demonstrate an error detection scheme that does not employ ancilla qubits or mid-circuit measurements. This is achieved by expanding the Hilbert space where a single logical qubit is encoded using several physical qubits. For example, one possible two qubit encoding identifies ∣0⟩L=∣01⟩|0\rangle_L=|01\rangle∣0⟩L​=∣01⟩ and ∣1⟩L=∣10⟩|1\rangle_L=|10\rangle∣1⟩L​=∣10⟩. If during the final measurement a ∣11⟩|11\rangle∣11⟩ or ∣00⟩|00\rangle∣00⟩ is observed an error is declared and the run is not included in subsequent analysis. We provide codewords for a simple bit-flip encoding, a way to encode the states, a way to implement logical U3U_3U3​ and logical CxC_xCx​ gates, and a description of which errors can be detected. We then run Greenberger-Horne-Zeilinger circuits on the transmon based IBM quantum computers, with an input space of N∈{2,3,4,5}N\in\{2,3,4,5\}N∈{2,3,4,5} logical qubits and Q∈{1,2,3,4,5}Q\in\{1,2,3,4,5\}Q∈{1,2,3,4,5} physical qubits per logical qubit. The results are compared relative to Q=1Q=1Q=1 with and without error detection and we find a significant improvement for Q∈{2,3,4}Q\in\{2,3,4\}Q∈{2,3,4}.
Fighting against noise is crucial for Noisy Intermediate Scale Quantum (NISQ) devices to demonstrate practical quantum applications. In this work, we give a paradigm of quantum error mitigation based on the vectorization of density matrices for variational quantum algorithms. Different from the ideas of existing quantum error mitigation methods that try to distill noiseless information from noisy quantum states, our proposal directly changes the way of encoding information and maps the density matrices of noisy quantum states to noiseless virtual pure states, which is realized by an NISQ-friendly measurement protocol and a classical postprocessing procedure. Our protocol requires no knowledge of the noise model, no ability to tune the noise strength, and no ancilla qubits for complicated controlled unitaries. Under our encoding, NISQ devices are always preparing virtual pure quantum states that are highly desired resources for variational quantum algorithms to have good performance in many tasks. We show how this protocol can be well fitted into variational quantum algorithms. We give several concrete ansatz constructions that are suitable for our proposal and do theoretical analysis on the sampling complexity, the expressibility, and the trainability. We also give a discussion on how this protocol is influenced by large noise and how it can be well combined with other quantum error mitigation protocols. The effectiveness of our proposal is demonstrated by various numerical experiments.
Near-term quantum computers have been built as intermediate-scale quantum devices and are fragile against quantum noise effects, namely, NISQ devices. Traditional quantum-error-correcting codes are not implemented on such devices and to perform quantum computation in good accuracy with these machines we need to develop alternative approaches for mitigating quantum computational errors. In this work, we propose quantum error mitigation (QEM) scheme for quantum computational errors which occur due to couplings with environments during gate operations, i.e., decoherence. To establish our QEM scheme, first we estimate the quantum noise effects on single-qubit states and represent them as groups of quantum circuits, namely, quantum-noise-effect circuit groups. Then our QEM scheme is conducted by subtracting expectation values generated by the quantum-noise-effect circuit groups from those obtained by the quantum circuits for the quantum algorithms under consideration. As a result, the quantum noise effects are reduced, and we obtain approximately the ideal expectation values via the quantum-noise-effect circuit groups and the numbers of elementary quantum circuits composing them scale polynomial with respect to the products of the depths of quantum algorithms and the numbers of register bits. To numerically demonstrate the validity of our QEM scheme, we run noisy quantum simulations of qubits under amplitude damping effects for four types of quantum algorithms. Furthermore, we implement our QEM scheme on IBM Q Experience processors and examine its efficacy. Consequently, the validity of our scheme is verified via both the quantum simulations and the quantum computations on the real quantum devices. Our QEM scheme is solely composed of quantum-computational operations (quantum gates and measurements), and thus, it can be conducted by any type of quantum device. In addition, it can be applied to error mitigation for many other types of quantum noise effects as well as noisy quantum computing of long-depth quantum algorithms.
Correcting errors due to noise in quantum circuits run on current and near-term quantum hardware is essential for any convincing demonstration of quantum advantage. Indeed, in many cases it has been shown that noise renders quantum circuits efficiently classically simulable, thereby destroying any quantum advantage potentially offered by an ideal (noiseless) implementation of these circuits. Although the technique of quantum error correction (QEC) allows correcting these errors very accurately, QEC usually requires a large overhead of physical qubits which is not reachable with currently available quantum hardware. This has been the motivation behind the field of quantum error mitigation, which aims at developing techniques to correct an important part of the errors in quantum circuits, while also being compatible with current and near-term quantum hardware. In this work, we present a technique for quantum error mitigation which is based on a technique from quantum verification, the so-called accreditation protocol, together with postselection. Our technique allows for correcting the expectation value of an observable OOO, which is the output of multiple runs of noisy quantum circuits, where the noise in these circuits is at the level of preparations, gates, and measurements. We discuss the sample complexity of our procedure and provide rigorous guarantees of errors being mitigated under some realistic assumptions on the noise. Our technique is tailored to time-dependent behaviors, as we allow for the output states to be different between different runs of the accreditation protocol. We validate our findings by running our technique on currently available quantum hardware.