Measuring All Compatible Operators in One Series of Single-Qubit Measurements Using Unitary Transformations

Yen, Tzu-Ching; Verteletskyi, Vladyslav; Izmaylov, Artur F.

doi:10.1021/acs.jctc.0c00008

Cited by 144 publications

(162 citation statements)

References 46 publications

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“…The costs and errors owing to the measurements can be reduced using simultaneous measurements and/or other techniques. Such techniques have been actively developed [65][66][67][68][69][70][71][72][73][74][75], and they can be used to alleviate the problems caused by measuring 1 and 2RDMs. In practice, OO-VQE repeatedly performs the VQE and the orbital optimization until convergence.…”

Section: Orbital Optimized Unitary Coupled Cluster Doublesmentioning

confidence: 99%

Orbital optimized unitary coupled cluster theory for quantum computer

Mizukami

Mitarai

Nakagawa³

et al. 2020

Phys. Rev. Research

108

106

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We propose an orbital optimized method for unitary coupled cluster theory (OO-UCC) within the variational quantum eigensolver (VQE) framework for quantum computers. OO-UCC variationally determines the coupled cluster amplitudes and also molecular orbital coefficients. Owing to its fully variational nature, first-order properties are readily available. This feature allows the optimization of molecular structures in VQE without solving any additional equations. Furthermore, the method requires smaller active space and shallower quantum circuits than UCC to achieve the same accuracy. We present numerical examples of OO-UCC using quantum simulators, which include the geometry optimization of water and ammonia molecules using analytical first derivatives of the VQE.

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Section: Orbital Optimized Unitary Coupled Cluster Doublesmentioning

confidence: 99%

Orbital optimized unitary coupled cluster theory for quantum computer

Mizukami

Mitarai

Nakagawa³

et al. 2020

Phys. Rev. Research

108

106

View full text Add to dashboard Cite

show abstract

“…Recently, a series of papers [16][17][18][19][20][21][22] * have appeared that make progress on both the collecting strategy and rotation circuit construction problems. Our paper is in the same arena and addresses both problems.…”

Section: Introductionmentioning

confidence: 99%

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

Crawford

Straaten

Wang

et al. 2021

Quantum

145

209

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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 SORTED INSERTION, 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 kn−k(k+1)/2 and O(kn/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, SORTED INSERTION outperforms four conventional greedy colouring algorithms that seek the minimum number of collections.

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“…Recent work has focused on mapping this problem to that of clique finding or coloring of a graph [20], and applying approximate algorithms to these known NP-hard problems [21]. These schemes achieves constant or empirically determined linear scaling improvements over an approach that measures each term individually [20,[22][23][24][25]. However, the commutation relations between local qubit or local fermionic operators has significant regularity not utilized in naive graph-theoretic algorithms.…”

Section: Introductionmentioning

confidence: 99%

Nearly Optimal Measurement Scheduling for Partial Tomography of Quantum States

2020

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Many applications of quantum simulation require one to prepare and then characterize quantum states by efficiently estimating k-body reduced density matrices (k-RDMs), from which observables of interest may be obtained. For instance, the fermionic 2-RDM contains the energy, charge density, and energy gradients of an electronic system, while the qubit 2-RDM contains the spatial correlation functions of magnetic systems. Naive estimation of such RDMs requires repeated state preparations for each matrix element, which makes for prohibitively large computation times. However, commuting matrix elements may be measured simultaneously, allowing for a significant cost reduction. In this work, we design schemes for such a parallelization with near-optimal complexity in the system size N. We first describe a scheme to sample all elements of a qubit k-RDM using only Oð3 k log k−1 NÞ unique measurement circuits, an exponential improvement over prior art. We then describe a scheme for sampling all elements of the fermionic 2-RDM using only OðN 2 Þ unique measurement circuits, each of which requires only a local OðNÞ-depth measurement circuit. We prove a lower bound of Ωðϵ −2 N k Þ on the number of state preparations, Clifford circuits, and measurement in the computational basis required to estimate all elements of a fermionic k-RDM, making our scheme for sampling the fermionic 2-RDM asymptotically optimal. We finally construct circuits to sample the expectation value of a linear combination of ω anticommuting two-body fermionic operators with only OðωÞ gates on a linear array. These circuits allows for sampling any linear combination of fermionic 2-RDM elements in OðN 4 =ωÞ time, with a significantly lower measurement circuit complexity than prior art. Our results improve the viability of near-term quantum simulation of molecules and strongly correlated material systems.

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Measuring All Compatible Operators in One Series of Single-Qubit Measurements Using Unitary Transformations

Cited by 144 publications

References 46 publications

Orbital optimized unitary coupled cluster theory for quantum computer

Orbital optimized unitary coupled cluster theory for quantum computer

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

Nearly Optimal Measurement Scheduling for Partial Tomography of Quantum States

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