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 $N^{\frac{2}{3}}$ with the number of Pauli strings $N$ 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.
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 N32 with the number of Pauli strings N 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.
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.
We introduce an approach for estimating the expectation values of arbitrary n-qubit matrices M∈C2n×2n on a quantum computer. In contrast to conventional methods like the Pauli decomposition that utilize 4n distinct quantum circuits for this task, our technique employs at most 2n 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 2n 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.
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.
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(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 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.
In this work, we consider one of the prototypical tasks for characterizing the structure of noise in quantum devices: estimating eigenvalues of an n -qubit Pauli noise channel. Prior work [Chen , Phys. Rev. A 105, 032435 (2022)] has proved no-go theorems for this task in the practical regime in which one has a limited amount of quantum memory; i.e., any protocol with ≤ 0.99 n ancilla qubits of quantum memory must make exponentially many measurements, provided that it is . Such protocols can only interact with the channel by repeatedly preparing a state, passing it through the channel, and measuring immediately afterward. Surprisingly, in this work we show that protocols with an extremely small amount of quantum memory achieve an exponential advantage for this task. We give a protocol that can estimate any prescribed set of eigenvalues A of a Pauli channel to error ε using only O ( log log ( | A | ) / ε 2 ) ancilla qubits, O ~ ( n 2 / ε 2 ) measurements, and O ( n 2 | A | / ε 2 ) queries to the Pauli channel. In contrast, we show that any protocol with zero ancilla—even a concatenating one—must make Ω ( 2 n / ε 2 ) measurements. We also prove that the number of queries cannot be improved significantly: with k ancilla qubits, Ω ( 2 ( n − k ) / 3 ) queries to the channel are necessary to learn all eigenvalues, even for concatenating protocols.
Machine learning provides a scalable solution for quantum error mitigation. However, the selection of appropriate Pauli strings for inclusion in training data remains a challenge. Current methods rely on heuristic or uniform random sampling, requiring data for every Pauli string in the Hamiltonian, a process that scales linearly with measurements and grows with system size. To address this, we introduce quantum error mitigation with prior knowledge of Pauli weights (Pauli weight quantum error mitigation (Pi-QEM)), a systematic framework that selects training observables based on Pauli weight. By leveraging the relationship between variance and locality in parameterized quantum circuits, Pi-QEM trains on a small subset of dominant, low-weight Pauli strings. In numerical simulations of molecular systems on a noisy IBM quantum backend, Pi-QEM reduces ground-state energy estimation error by up to 34.01% using just a single dominant local observable, offering an efficient, scalable pathway for high-precision error mitigation on NISQ devices.