Numerical Error Extraction by Quantum Measurement Algorithm

TL;DR

The paper addresses the challenge of quantifying how gate errors decay with the convergence parameter in iterative quantum routines. It introduces NEEQMA, a measurement-based protocol that extracts the constants of the convergence law by fitting a parametric error model to observables measured on a quantum processor. The method is demonstrated on Lie-Trotter Hamiltonian simulation and quantum signal processing-based eigenvalue filtering, yielding concrete constants that enable predicting gate error for higher-order approximations and thus selecting the smallest necessary convergence parameters. This enables more efficient quantum circuits by tying hardware measurements directly to algorithmic accuracy requirements.

Abstract

Important quantum algorithm routines allow the implementation of specific quantum operations (a.k.a. gates) by combining basic quantum circuits with an iterative structure. In this structure, the number of repetitions of the basic circuit pattern is associated to convergence parameters. This iterative structure behaves similarly to function approximation by series expansion: the higher the truncation order, the better the target gate (i.e. operation) approximation. The asymptotic convergence of the gate error with respect to the number of basic pattern repetitions is known. It is referred to as the query complexity. The underlying convergence law is bounded, but not in an explicit fashion. Upper bounds are generally too pessimistic to be useful in practice. The actual convergence law contains constants that depend on the joint properties of the matrix encoded by the query and the initial state vector, which are difficult to compute classically. This paper proposes a strategy to study this convergence law and extract the associated constants from the gate (operation) approximation at different accuracy (convergence parameter) constructed directly on a Quantum Processing Unit (QPU). This protocol is called Numerical Error Extraction by Quantum Measurement Algorithm (NEEQMA). NEEQMA concepts are tested on specific instances of Quantum Signal Processing (QSP) and Hamiltonian Simulation by Trotterization. Knowing theexact convergence constants allows for selecting the smallest convergence parameters that enable reaching the required gate approximation accuracy, hence satisfying the quantum algorithm's requirements.

Figures (15)

• Figure 1: $\mathrm{sign} \circ \cos(\pi x)$ function polynomial approximations at different orders $d$. The top graph represents the function approximations evaluated at many values in the $[ -1, 1 ]$ interval. It shows that the error fluctuates depending on the evaluation point. The bottom curve shows how the approximations converge as $d$ increases for a chosen value equals to $x = 0.56$ chosen for illustrative purposes; the associated error is a scalar. The purple vertical lines correspond to the eigenvalues associated with the largest $\alpha$ values of \ref{['section_qsp']} experiment.
• Figure 2: neeqma Workflow.
• Figure 3: Trotter plus Hadamard-test quantum circuit. Here: $\widehat{H_{p}} = \widehat{H_{1}} + \widehat{H_{2}}$. The dashed box delimits the Trotterization routine. The $\widehat{S}^{\dag}$ gate is used to measure the imaginary part (but replaced by an identity gate when measuring the real part).
• Figure 4: Observable measurements (real and imaginary part of the Trotterized Hamiltonian simulation) at different Trotter numbers $n$ with the associated error model, and with the free-parameters adjusted by classical optimization. This curve was computed with a Hamiltonian simulation time $t = 1$.
• Figure 5: Error model with the convergence law constants obtained from the quantum circuit modelling and directly from matrix calculation (using numpy) with respect to the convergence parameter $n$ at the Hamiltonian simulation time $t = 1$. Note that the increasing distance between the two curves is an artefact of the graph's log scale.