The dominant eigenvector of a noisy quantum state

Abstract

Although near-term quantum devices have no comprehensive solution for correcting errors, numerous techniques have been proposed for achieving practical value. Two works have recently introduced the very promising error suppression by derangements (ESD) and virtual distillation (VD) techniques. The approach exponentially suppresses errors and ultimately allows one to measure expectation values in the pure state as the dominant eigenvector of the noisy quantum state. Interestingly this dominant eigenvector is, however, different than the ideal computational state and it is the aim of the present work to comprehensively explore the following fundamental question: how significantly different are these two pure states? The motivation for this work is two-fold. First, comprehensively understanding the effect of this coherent mismatch is of fundamental importance for the successful exploitation of noisy quantum devices. As such, the present work rigorously establishes that in practically relevant scenarios the coherent mismatch is exponentially less severe than the incoherent decay of the fidelity—where the latter can be suppressed exponentially via the ESD/VD technique. Second, the above question is closely related to central problems in mathematics, such as bounding eigenvalues of a sum of two matrices (Weyl inequalities)—solving of which was a major breakthrough. The present work can be viewed as a first step towards extending the Weyl inequalities to eigenvectors of a sum of two matrices—and completely resolves this problem for the special case of the considered density matrices.

Shadow Distillation: Quantum Error Mitigation with Classical Shadows for Near-Term Quantum Processors

“…Note that the second-order mitigation (m = 2) has its limitations and even with infinitely many measurements one cannot completely eliminate the errors. One can obtain the full density matrix by taking the average of the measurement snapshots, which allows mitigation with an arbitrary m. Therefore, the ultimate mitigation (m → ∞) can be achieved by obtaining the dominant eigenvector of ρ [38], which takes the time O(2 3nq ). However, taking the latter approach has the same complexity as simulating the full quantum system and is unlikely to be useful beyond a proof-of-concept illustration.…”

“…The residual errors in Fig. 4 either correspond to higher-order incoherent errors, incoherent errors that modify the eigenvectors of ρ (also known as the coherent mismatch [12,14,38]), or coherent errors originating from under(over)-rotation in two-qubit gates, which is a known source of error in trapped-ion systems [42].…”

Multicore Quantum Computing

“…[11], if we entangle these copies with the derangement circuit and estimate the probability Prob 0 that the ancilla qubit collapses into state |0 , we can formally obtain the expectation value Tr[ρ n σ]/Tr[ρ n ]. This allows us to suppress errors in estimating the expectation value of an observable σ exponentially in n. The main limitation of the approach is that a small coherent mismatch in the dominant eigenvector of the state ρ may ultimately bias our estimates, however, this mismatch is exponentially less severe than the incoherent decay of the fidelity [76].…”

Quantum Error Mitigation Relying on Permutation Filtering

“…where |ψ i denotes the eigenvector associated with the i-th largest eigenvalue of ρ, and |ψ = |ψ 1 is the dominant eigenvector, which approximates the noise-free output state [31], [38]. Inspired by these observations, Koczor [31] proposed the permutation-based quantum error mitigation concept, as portrayed in Fig.…”

“…Closer scrutiny reveals that the performance of both the Type-1 and Type-2 third-order filters become similar when the number of stages is large, especially when it is larger than 45. By contrast, the performance of the Type-2 filter is near-optimal when the number of stages is moderate (around [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40]. This trend may prevail, because the Pareto fit becomes more accurate, when the noise bandwidth is moderate.…”