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arXiv:1908.06942 · Quantum

Efficient quantum measurement of Pauli operators in the presence of finite sampling error

Ophelia Crawford, Barnaby van Straaten, Daochen Wang, Thomas Parks, Earl T. Campbell, STEPHEN D. BRIERLEY

Published Jan 20, 2021• 183 citations

Abstract

Estimating the expectation value of an operator corresponding to an observable is a fundamental task in quantum computation. It is often impossible to obtain such estimates directly, as the computer is restricted to measuring in a fixed computational basis. One common solution splits the operator into a weighted sum of Pauli operators and measures each separately, at the cost of many measurements. An improved version collects mutually commuting Pauli operators together before measuring all operators within a collection simultaneously. The effectiveness of doing this depends on two factors. Firstly, we must understand the improvement offered by a given arrangement of Paulis in collections. In our work, we propose two natural metrics for quantifying this, operating under the assumption that measurements are distributed optimally among collections so as to minimise the overall finite sampling error. Motivated by the mathematical form of these metrics, we introduce S ORTED I NSERTION, a collecting strategy that exploits the weighting of each Pauli operator in the overall sum. Secondly, to measure all Pauli operators within a collection simultaneously, a circuit is required to rotate them to the computational basis. In our work, we present two efficient circuit constructions that suitably rotate any collection of k independent commuting n -qubit Pauli operators using at most k n − k ( k + 1 ) / 2 and O ( k n / log ⁡ k ) two-qubit gates respectively. Our methods are numerically illustrated in the context of the Variational Quantum Eigensolver, where the operators in question are molecular Hamiltonians. As measured by our metrics, S ORTED I NSERTION outperforms four conventional greedy colouring algorithms that seek the minimum number of collections.

DOI: 10.22331/q-2021-01-20-385

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Estimating the expectation value of an operator corresponding to an observable is a fundamental task in quantum computation. It is often impossible to obtain such estimates directly, as the computer is restricted to measuring in a fixed computational basis. One common solution splits the operator into a weighted sum of Pauli operators and measures each separately, at the cost of many measurements. An improved version collects mutually commuting Pauli operators together before measuring all operators within a collection simultaneously. The effectiveness of doing this depends on two factors. Firstly, we must understand the improvement offered by a given arrangement of Paulis in collections. In our work, we propose two natural metrics for quantifying this, operating under the assumption that measurements are distributed optimally among collections so as to minimise the overall finite sampling error. Motivated by the mathematical form of these metrics, we introduce S ORTED I NSERTION, a collecting strategy that exploits the weighting of each Pauli operator in the overall sum. Secondly, to measure all Pauli operators within a collection simultaneously, a circuit is required to rotate them to the computational basis. In our work, we present two efficient circuit constructions that suitably rotate any collection of k independent commuting n -qubit Pauli operators using at most k n − k ( k + 1 ) / 2 and O ( k n / log ⁡ k ) two-qubit gates respectively. Our methods are numerically illustrated in the context of the Variational Quantum Eigensolver, where the operators in question are molecular Hamiltonians. As measured by our metrics, S ORTED I NSERTION outperforms four conventional greedy colouring algorithms that seek the minimum number of collections.

Measuring quantum observables by grouping terms that can be rotated to sums of only products of Pauli ˆz operators (Ising form) is proven to be efficient in near-term quantum computing algorithms.This approach requires extra unitary transformations to rotate the state of interest so that the measurement of a fragment’s Ising form would be equivalent to measurement of the fragment for the unrotated state. These extra rotations allow one to perform a fewer number of measurements by grouping more terms into the measurable fragments with a lower overall estimator variance. However, previous estimations of the number of measurements did not take into account non-unit fidelity of quantum gates implementing the additional transformations. Through a circuit fidelity reduction, additional transformations introduce extra uncertainty and increase the needed number of measurements. Here we consider a simple model for errors introduced by additional gates needed in schemes involving grouping of commuting Pauli products. For a set of molecular electronic Hamiltonians, we confirm that the numbers of measurements in schemes using non-local qubit rotations are still lower than those in their local qubit rotation counterparts, even after accounting for uncertainties introduced by additional gates.

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.

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

{\mathrm{H}}_{2}

using fewer measurements than is possible with single-qubit gates with no statistically significant difference in accuracy.

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

n

-qubit molecular Hamiltonians in quantum chemistry calculations, it is necessary to evaluate

O(n^4)

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

n

-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

n

-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.

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.

The Monte Carlo Projective Quantum Eigensolver (MC-PQE) was recently introduced as an alternative to hybrid algorithms like the variational quantum eigensolver (VQE). By using a quantum Monte Carlo-inspired scheme for energy estimation, MC\nobreakdash-PQE was found to decrease the required measurement cost relative to comparable conventional approaches. However, in the context of VQE, numerous techniques have been developed to reduce the measurement overhead by joint measurement of multiple observables. In this work, we extend these approaches to the asymmetric expectation values required in MC-PQE and assess the performance of different techniques for various molecular systems of up to 12 qubits. We find full commuting Pauli term grouping combined with tailored measurement allocation techniques leads to a 5-10

\times

reduction in standard error for the same total number of quantum measurements. Conventional Hamiltonian grouped measurement outperforms tractable classical shadows tomography based techniques for the considered systems.