Exponential Error Suppression for Near-Term Quantum Devices

Abstract

As quantum computers mature, quantum error correcting codes (QECs) will be adopted in order to suppress errors to any desired level E at a cost in qubit-count n that is merely poly-logarithmic in E −1 . However in the NISQ era, the complexity and scale required to adopt even the smallest QEC is prohibitive. Instead, error mitigation techniques have been employed; typically these do not require a significant increase in qubit-count but cannot provide exponential error suppression. 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. At the cost of a qubit count increase by a factor of n ≥ 2 the error becomes Q n where Q < 1 is a suppression factor that depends on the entropy of the error probabilities. The approach takes n independently-prepared circuit outputs and applies a controlled derangement operator to create a state whose symmetries prevent error-burdened states from contributing bias to the expected value. The approach is therefore 'NISQ-friendly' as it is modular in the main computation and requires only a shallow circuit that bridges the n copies immediately prior to measurement. Imperfections in our derangement circuit do degrade performance and therefore we propose an approach to efficiently mitigate this effect to arbitrary precision due to the remarkable properties of derangements. a) they decompose into a linear number of elementary gates -limiting the impact of noise b) they are highly resilient to noise and the effect of imperfections on them is (almost) trivial. In numerical simulations validating our approach we confirm error suppression below 10 −6 for circuits consisting of several hundred noisy gates (two-qubit gate error 0.5%) using no more than n = 4 circuit copies.

The dominant eigenvector of a noisy quantum state

“…Quantum circuit of a possible implementation of the ESD/VD error suppression approach [23,24]. Reproduced from [23].…”

“…Quantum circuit of a possible implementation of the ESD/VD error suppression approach [23,24]. Reproduced from [23]. CC BY 4.0. n copies of the noisy quantum state ρ are prepared and entangled via a controlled-derangement operator D n -a generalisation of the SWAP operation that permutes the n quantum registers.…”

“…While reference [24] mostly focuses on the n = 2 scenario and proposes a resource efficient variant that does not require an ancilla qubit when n = 2, reference [23] presents explicit constructions of the approach for n 2. A possible implementation is illustrated in figure 2 which uses a controlled-derangement operation and allows one to measure expectation values of the form Tr[ρ n O]/Tr[ρ n ] with respect to an observable O.…”

“…Given a noisy quantum state ρ, how well does its dominant eigenvector approximate the state that one would obtain from a perfect, noise-free computation? It was already noted in references [23,24] that even incoherent errors will in general introduce a drift in the dominant eigenvector. This drift was named 'coherent mismatch' and 'noise floor' by the two works.…”

“…These typically aim to learn the effect of imperfections on expectation values of observables and try to predict their ideal, noise-free values. A very promising approach has recently been introduced by two independent works and was named error suppression by derangements (ESD [23]) and virtual distillation (VD [24]). This technique prepares n copies of the noisy quantum state and in turn allows to suppress errors in expectation values exponentially when increasing n. The approach relies on the assumption that the dominant Figure 1.…”