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arXiv:2411.04240

Randomly Compiled Quantum Simulation with Exponentially Reduced Circuit Depths

James D. Watson

Published Nov 6, 2024• 1 citation
arXivHTMLPDF

Abstract

The quantum stochastic drift protocol, also known as qDRIFT, has become a popular algorithm for implementing time-evolution of quantum systems using randomised compiling. In this work we develop qFLO, a higher order randomised algorithm for time-evolution. To estimate an observable expectation value at time $T$ to precision $ε$, we show it is sufficient to use circuit depths of $O(T^2\log(1/ε))$ -- an exponential improvement over standard qDRIFT requirements with respect to $ε$. The protocol achieves this using $O(1/ε^2)$ repeated runs of the standard qDRIFT protocol combined with classical post-processing in the form of Richardson extrapolation. Notably, it requires no ancillary qubits or additional control gates making it especially promising for near-term quantum devices. Furthermore, it is well-conditioned and inherits many desirable properties of randomly compiled simulation methods, including circuit depths that do not explicitly depend on the number of terms in the Hamiltonian.

DOI
10.48550/arxiv.2411.04240

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The quantum stochastic drift protocol, also known as qDRIFT, has become a popular algorithm for implementing time-evolution of quantum systems using randomised compiling. In this work we develop qFLO, a higher order randomised algorithm for time-evolution. To estimate an observable expectation value at time TTT to precision εεε, we show it is sufficient to use circuit depths of O(T2log⁡(1/ε))O(T^2\log(1/ε))O(T2log(1/ε)) -- an exponential improvement over standard qDRIFT requirements with respect to εεε. The protocol achieves this using O(1/ε2)O(1/ε^2)O(1/ε2) repeated runs of the standard qDRIFT protocol combined with classical post-processing in the form of Richardson extrapolation. Notably, it requires no ancillary qubits or additional control gates making it especially promising for near-term quantum devices. Furthermore, it is well-conditioned and inherits many desirable properties of randomly compiled simulation methods, including circuit depths that do not explicitly depend on the number of terms in the Hamiltonian.
Abstract Quantum phase estimation (QPE) is a fundamental tool in quantum computing, facilitating efficient simulations of complex problems in quantum chemistry and materials science. While most phase estimation algorithms are deterministic, recent advancements indicate that incorporating randomness can enhance performance. This study introduces a framework for randomized QPE that merges the benefits of randomized compilation with phase estimation algorithms based on quantum signal processing. Our proposed algorithms effectively reduce circuit depths by eliminating the need for precise Hamiltonian time evolution, making them advantageous for digital quantum computers estimating the eigenvalue and eigenvector properties of Hamiltonians. Notably, our findings show that the quantum stochastic drift protocol (qDRIFT)-based randomized algorithm surpasses the original phase estimation with qDRIFT, especially in scaling inverse failure probabilities. We also establish that a circuit depth of O ( log ⁡ ( M ) ) suffices for estimating M distinct observables. The protocol is executed through multiple iterations of the randomized algorithms combined with classical shadow techniques. Overall, our framework retains many advantages of the randomized compilation technique, making it a compelling solution for challenges in quantum chemistry.
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Hamiltonian simulation is known to be one of the fundamental building blocks of a variety of quantum algorithms such as its most immediate application, that of simulating many-body systems to extract their physical properties. In this work, we present qSWIFT, a high-order randomized algorithm for Hamiltonian simulation. In qSWIFT, the required number of gates for a given precision is independent of the number of terms in the Hamiltonian, while the systematic error is exponentially reduced with regard to the order parameter. In this respect, our qSWIFT is a higher-order counterpart of the previously proposed quantum stochastic drift protocol (qDRIFT), the number of gates in which scales linearly with the inverse of the precision required. We construct the qSWIFT channel and establish a rigorous bound for the systematic error quantified by the diamond norm. qSWIFT provides an algorithm to estimate given physical quantities by using a system with one ancilla qubit, which is as simple as other product-formula-based approaches such as regular Trotter-Suzuki decompositions and qDRIFT. Our numerical experiment reveals that the required number of gates in qSWIFT is significantly reduced compared to qDRIFT. In particular, the advantage is significant for problems where high precision is required; e.g., to achieve a systematic relative propagation error of 10−6 , the required number of gates in third-order qSWIFT is 1000 times smaller than that of qDRIFT. Published by the American Physical Society 2024
Randomised quantum simulation algorithms such as qDRIFT reduce circuit depth by replacing deterministic product formulas with stochastic compilations, but achieving high accuracy can require many steps. In vanilla qDRIFT, Hamiltonian terms are sampled with probabilities proportional to their coefficient magnitudes. We propose StateDRIFT, a variant of qDRIFT which optimises its sampling distribution to estimate time-evolved quantities using knowledge of the initial state and target observable, with a provable truncation bound that keeps qDRIFT's guarantees. The distribution at each depth is chosen by solving an optimisation problem targeting bias and/or variance. StateDRIFT can significantly reduce the bias and variance of estimators, and hence circuit depth, which we test on short-time dynamics of lattice models and molecules under Jordan-Wigner up to 18 qubits. For example, with BeH2 StateDRIFT can reduce the number of two-qubit gates required to achieve a given accuracy from 11,201 to 983. StateDRIFT involves measuring observables and commutators on the initial state, so we propose two practical ways to estimate these: classical surrogates such as tensor networks, and classical shadows, which recover most or all of the saving from exact oracles in the systems tested.
Randomized product formulas provide simple and flexible methods for quantum simulation. While qDRIFT-type randomized simulation has been extended to Lindbladian dynamics, higher-order randomized compilers have primarily been developed for Hamiltonian simulation. In this work, we develop first- and second-order randomized sampling algorithms for Lindbladian simulation in both the time-independent and time-dependent settings. Building on first-order qDRIFT-type Lindbladian simulation, we introduce a qSWIFT-type second-order correction for time-independent Lindbladians with a finite local decomposition. For λt≥1λt \geq 1λt≥1, where λλλ is the sum of the Hamiltonian coefficients and dissipative rates, the resulting approximation achieves normalized diamond-norm error at most ε\varepsilonε using r=O(λ2t2/ε)r=O(λ^2 t^2/\sqrt{\varepsilon})r=O(λ2t2/ε​) time slices, improving the ε−1\varepsilon^{-1}ε−1 precision dependence of the first-order qDRIFT-type approximation. We then develop first- and second-order continuous-time sampling algorithms for general time-dependent Lindbladians, with the second-order construction likewise achieving an ε−1/2\varepsilon^{-1/2}ε−1/2 dependence on the target precision in the number of time slices. We further extend the time-dependent construction to a linear-combination input model over local Hamiltonian and dissipative generators, allowing the randomized sampler to be formulated in terms of local evolutions rather than the full time-dependent Lindbladian. Because the second-order corrected maps need not be CPTP, we use them to estimate observable expectation values rather than to deterministically prepare output states.
Simulation of quantum chemistry is expected to be a principal application of quantum computing. In quantum simulation, a complicated Hamiltonian describing the dynamics of a quantum system is decomposed into its constituent terms, where the effect of each term during time-evolution is individually computed. For many physical systems, the Hamiltonian has a large number of terms, constraining the scalability of established simulation methods. To address this limitation we introduce a new scheme that approximates the actual Hamiltonian with a sparser Hamiltonian containing fewer terms. By stochastically sparsifying weaker Hamiltonian terms, we benefit from a quadratic suppression of errors relative to deterministic approaches. Relying on optimality conditions from convex optimisation theory, we derive an appropriate probability distribution for the weaker Hamiltonian terms, and compare its error bounds with other probability ansatzes for some electronic structure Hamiltonians. Tuning the sparsity of our approximate Hamiltonians allows our scheme to interpolate between two recent random compilers: qDRIFT and randomized first order Trotter. Our scheme is thus an algorithm that combines the strengths of randomised Trotterisation with the efficiency of qDRIFT, and for intermediate gate budgets, outperforms both of these prior methods.