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arXiv:2401.17640

Towards enhancing quantum expectation estimation of matrices through partial Pauli decomposition techniques and post-processing

Dingjie Lu, Yangfan Li, Dax Enshan Koh, Zhao Wang, Jun Liu, Zhuangjian Liu

Published Jan 31, 2024• 0 citations
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Abstract

We introduce an approach for estimating the expectation values of arbitrary $n$-qubit matrices $M \in \mathbb{C}^{2^n\times 2^n}$ on a quantum computer. In contrast to conventional methods like the Pauli decomposition that utilize $4^n$ distinct quantum circuits for this task, our technique employs at most $2^n$ unique circuits, with even fewer required for matrices with limited bandwidth. Termed the \textit{partial Pauli decomposition}, our method involves observables formed as the Kronecker product of a single-qubit Pauli operator and orthogonal projections onto the computational basis. By measuring each such observable, one can simultaneously glean information about $2^n$ distinct entries of $M$ through post-processing of the measurement counts. This reduction in quantum resources is especially crucial in the current noisy intermediate-scale quantum era, offering the potential to accelerate quantum algorithms that rely heavily on expectation estimation, such as the variational quantum eigensolver and the quantum approximate optimization algorithm.

DOI
10.48550/arxiv.2401.17640

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We introduce an approach for estimating the expectation values of arbitrary nnn-qubit matrices M∈C2n×2nM \in \mathbb{C}^{2^n\times 2^n}M∈C2n×2n on a quantum computer. In contrast to conventional methods like the Pauli decomposition that utilize 4n4^n4n distinct quantum circuits for this task, our technique employs at most 2n2^n2n unique circuits, with even fewer required for matrices with limited bandwidth. Termed the \textit{partial Pauli decomposition}, our method involves observables formed as the Kronecker product of a single-qubit Pauli operator and orthogonal projections onto the computational basis. By measuring each such observable, one can simultaneously glean information about 2n2^n2n distinct entries of MMM through post-processing of the measurement counts. This reduction in quantum resources is especially crucial in the current noisy intermediate-scale quantum era, offering the potential to accelerate quantum algorithms that rely heavily on expectation estimation, such as the variational quantum eigensolver and the quantum approximate optimization algorithm.
Expectation value estimation is ubiquitous in quantum algorithms. The expectation value of a Hamiltonian, which is essential in various practical applications, is often estimated by measuring a large number of Pauli strings on quantum computers and performing classical post-processing. In the case of nnn-qubit molecular Hamiltonians in quantum chemistry calculations, it is necessary to evaluate O(n4)O(n^4)O(n4) Pauli strings, requiring a large number of measurements for accurate estimation. To reduce the measurement cost, we assess an existing idea that uses two copies of an nnn-qubit quantum state of interest and coherently measures them in the Bell basis, which enables the simultaneous estimation of the absolute values of expectation values of all the nnn-qubit Pauli strings. We numerically investigate the efficiency of energy estimation for molecular Hamiltonians of up to 12 qubits. The results show that, when the target precision is no smaller than tens of milli-Hartree, this method requires fewer measurements than conventional sampling methods. This suggests that the method may be useful for many applications that rely on expectation value estimation of Hamiltonians and other observables as well when moderate precision is sufficient.
The estimation of multi-qubit observables is a key task in quantum information science. The standard approach is to decompose a multi-qubit observable into a weighted sum of Pauli strings. The observable can then be estimated from projective single qubit measurements according to the Pauli strings followed by a classical summation. As the number of Pauli strings in the decomposition increases, shot-noise drastically builds up, and the accuracy of such estimation can be considerably compromised. Access to a single qubit quantum memory, where measurement data may be stored and accumulated can circumvent the build-up of shot noise. Here, we describe a many-qubit observable estimation approach to achieve this with a much lower number of interactions between the multi-qubit device and the single qubit memory compared to previous approaches. Our algorithm offers a reduction in the required number of measurements for a given target variance that scales N23N^{\frac{2}{3}}N32​ with the number of Pauli strings NNN in the observable decomposition. The low number of interactions between the multi-qubit device and the memory is desirable for noisy intermediate-scale quantum devices.
Estimating expectation values of large Hamiltonians is a central bottleneck in variational quantum algorithms such as the Variational Quantum Eigensolver. The standard approaches require decomposing the Hamiltonian into a sum of Pauli operators and estimating each term through separate measurement bases. This strategy relies on grouping commuting terms whose optimization is NP-hard. In this work, we propose and analyze a modification of variational algorithms by introducing an interferometric dynamics that enables the extraction of the expectation value [Formula: see text]H[Formula: see text] from a parametrized quantum circuit through repeated measurements of a single auxiliary qubit. Our scheme bypasses the need for Pauli term grouping by encoding the Hamiltonian information in interference fringes, enabling the direct estimation of global contributions. This interferometric approach is fully compatible with parametrized quantum circuits and integrates naturally with Variational Quantum Eigensolver.
The estimation of many-qubit observables is an essential task of quantum information processing. The generally applicable approach is to decompose the observables into weighted sums of multi-qubit Pauli strings, i.e., tensor products of single-qubit Pauli matrices, which can readily be measured with low-depth Clifford circuits. The accumulation of shot noise in this approach, however, severely limits the achievable variance for a finite number of measurements. We introduce a novel method, dubbed coherent Pauli summation (CPS), that circumvents this limitation by exploiting access to a single-qubit quantum memory in which measurement information can be stored and accumulated. CPS offers a reduction in the required number of measurements for a given variance that scales linearly with the number of Pauli strings in the decomposed observable. Our work demonstrates how a single long-coherence qubit memory can assist the operation of many-qubit quantum devices in a cardinal task.
Reducing the number of measurements required to estimate the expectation value of an observable is crucial for the variational quantum eigensolver to become competitive with state-of-the-art classical algorithms. To measure complicated observables such as a molecular electronic Hamiltonian, one of the common strategies is to partition the observable into linear combinations (fragments) of mutually commutative Pauli products. The total number of measurements for obtaining the expectation value is then proportional to the sum of variances of individual fragments. We propose a method that lowers individual fragment variances by modifying the fragments without changing the total observable expectation value. Our approach is based on adding Pauli products ("ghosts") that are compatible with members of multiple fragments. The total expectation value does not change because a sum of coefficients for each "ghost" Pauli product introduced to several fragments is zero. Yet, these additions change individual fragment variances because of the non-vanishing contributions of "ghost" Pauli products within each fragment. The proposed algorithm minimizes individual fragment variances using a classically efficient approximation of the quantum wavefunction for variance estimations. Numerical tests on a few molecular electronic Hamiltonian expectation values show several-fold reductions in the number of measurements in the "ghost" Pauli algorithm compared to those in the other recently developed techniques.
Measuring the expectation value of Pauli operators on prepared quantum states is a fundamental task in a multitude of quantum algorithms. Simultaneously measuring sets of Pauli operators allows for fewer measurements and an overall speedup of the measurement process. Using results from coloring random graphs, we give evidence for an upper bound on the expected minimum number of simultaneously measurable parts necessary to partition a set of Pauli operators. By expanding our gate set to allow multiqubit Clifford gates before measurement, we were able to cut down the number of measurements considerably compared to previous approaches. We conjecture that this improvement is, in general, linear with respect to the lengths of the Pauli operators. We conclude by demonstrating the feasibility of this approach by implementing the variational quantum eigensolver on the Aspen-4-4Q-A quantum processor unit. In doing so, we calculate the ground-state energies of H2{\mathrm{H}}_{2}H2​ using fewer measurements than is possible with single-qubit gates with no statistically significant difference in accuracy.