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arXiv:2212.07710 · Quantum

Quantum memory assisted observable estimation

Liubov A. Markovich, Attaallah Almasi, Sina Zeytinoǧlu, Johannes Borregaard

Published Mar 6, 2025• 0 citations
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Abstract

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.

DOI
10.22331/q-2025-03-06-1655

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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.
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.
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.
Accurately estimating expectation values of quantum observables with as few measurements as possible is crucial to many quantum computing applications. We introduce a framework that covers many of existing measurement strategies and introduce heuristics that can be used to enhance randomized schemes, including those based on Pauli grouping with inverse probability weighting and variants of the classical shadow algorithm. We show how to maximize information gain from such schemes, while carefully optimizing the distribution of possible measurements, and show that simple grouping algorithms can get close to, and in some cases exceed, state-of-the-art accuracy for unbiased estimation of expectation values on a standard quantum chemistry benchmark. We show how these randomized methods may be compared to more recent measurement schemes, such as shadow grouping, derandomized shadow, and overlapped grouping measurement, we show how the same strategies can be used to augment these schemes, and we demonstrate that we can reduce measurement costs by up to a factor of two by allowing Clifford measurement circuits for otherwise Clifford-less methods.
Generating samples from the output distribution of a quantum circuit is a ubiquitous task used as a building block of many quantum algorithms. Here we show how to accomplish this task on a noisy quantum processor lacking full-blown error correction for a special class of quantum circuits dominated by Clifford gates. Our approach is based on Coherent Pauli Checks (CPCs) that detect errors in a Clifford circuit by verifying commutation rules between random Pauli-type check operators and the considered circuit. Our main contributions are as follows. First, we derive a simple formula for the probability that a Clifford circuit protected by CPCs contains a logical error. In the limit of a large number of checks, the logical error probability is shown to approach the value ≈7εn/5{\approx}7εn/5≈7εn/5, where nnn is the number of qubits and εεε is the depolarizing error rate. Our formula agrees nearly perfectly with the numerical simulation results. Second, we show that CPCs are well-suited for quantum processors with a limited qubit connectivity. For example, the difference between all-to-all and linear qubit connectivity is only a 3X increase in the number of CNOT gates required to implement CPCs. Third, we describe simplified one-sided CPCs which are well-suited for mitigating measurement errors in the single-shot settings. Finally, we report an experimental demonstration of CPCs with up to 10 logical qubits and more than 100 logical CNOT gates. Our experimental results show that CPCs provide a marked improvement in the logical error probability for the considered task of sampling the output distribution of quantum circuits.
Generating samples from the output distribution of a quantum circuit is a ubiquitous task used as a building block of many quantum algorithms. Here we show how to accomplish this task on a noisy quantum processor lacking full-blown error correction for a special class of quantum circuits dominated by Clifford gates. Our approach is based on coherent Pauli checks (CPCs) that detect errors in a Clifford circuit by verifying commutation rules between random Pauli-type check operators and the considered circuit. Our main contributions are as follows. First, we derive a simple formula for the probability that a Clifford circuit protected by CPCs contains a logical error. In the limit of a large number of checks, the logical error probability is shown to approach the value $\ensuremath{\approx}7\ensuremath{\epsilon}n/5$, where nnn is the number of qubits and $\ensuremath{\epsilon}$ is the depolarizing error rate. Our formula agrees nearly perfectly with the numerical simulation results. Second, we show that CPCs are well suited for quantum processors with a limited qubit connectivity. For example, the difference between all-to-all and linear qubit connectivity is only a $3\ifmmode\times\else\texttimes\fi{}$ increase in the number of cnot gates required to implement CPCs. Third, we describe simplified one-sided CPCs, which are well suited for mitigating measurement errors in the single-shot settings. Finally, we report an experimental demonstration of CPCs with up to 10 logical qubits and more than 100 logical cnot gates. Our experimental results show that CPCs provide a marked improvement in the logical error probability for the considered task of sampling the output distribution of quantum circuits.
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.