Simulating Hydrodynamics on Noisy Intermediate-Scale Quantum Devices with Random Circuits

Abstract

In a recent milestone experiment, Google's processor Sycamore heralded the era of "quantum supremacy" by sampling from the output of (pseudo-)random circuits. We show that such random circuits provide tailor-made building blocks for simulating quantum many-body systems on noisy intermediate-scale quantum (NISQ) devices. Specifically, we propose an algorithm consisting of a random circuit followed by a trotterized Hamiltonian time evolution to study hydrodynamics and to extract transport coefficients in the linear response regime. We numerically demonstrate the algorithm by simulating the buildup of spatiotemporal correlation functions in one-and two-dimensional quantum spin systems, where we particularly scrutinize the inevitable impact of errors present in any realistic implementation. Importantly, we find that the hydrodynamic scaling of the correlations is highly robust with respect to the size of the Trotter step, which opens the door to reach nontrivial time scales with a small number of gates. While errors within the random circuit are shown to be irrelevant, we furthermore unveil that meaningful results can be obtained for noisy time evolutions with error rates achievable on near-term hardware. Our work emphasizes the practical relevance of random circuits on NISQ devices beyond the abstract sampling task.

Quantum versus classical dynamics in spin models: Chains, ladders, and square lattices

“…5 the decay of C (M) (t ) for the same values of and a fixed edge length L x = L y = 5. The overall situation appears to be similar to the one for the quasi-1D two-leg ladder, e.g., the relaxation is well described by a power law t −α with a diffusive exponent α, which is α = 1 in this 2D case [7]. For = 0.5 in Fig.…”

“…( 22) and the constant c is chosen in such a way that ρ r + c has nonnegative eigenvalues. Then, the correlation function can be rewritten as a standard expectation value [7,57,80],…”

“…Understanding the properties of quantum many-body systems out of equilibrium is a notoriously difficult task with relevance to various areas of modern physics, ranging from fundamental aspects of statistical mechanics [1,2] to more applied issues in material science and quantum information technology. Quantum spin systems are of particular importance in this context, since they describe the magnetism of certain compounds in nature [3], can be realized in new experimental platforms [4,5], or can be simulated on already available or future quantum computers [6,7].…”

Time-crystalline eigenstate order on a quantum processor

“…Still, a direct sampling of states is practically limited to small fractions of the computational basis (0.05% in this case) and would suffer from the exponential growth of the Hilbert space on larger systems. A more scalable alternative is to use random, highly entangled states to directly measure spectrally averaged quantities (quantum typicality 37 – 39 ; see Supplementary Information). The autocorrelator averaged over all bit strings agrees, up to an error exponentially small in , with , where ⟩ is a typical Haar-random many-body state in the Hilbert space of qubits.…”

Exploring Finite Temperature Properties of Materials with Quantum Computers

“…Here, we present an algorithm for generating canonical TPQ states on quantum computers with low quantum resource requirements, enabling the estimation of finite temperature properties of materials on NISQ devices. Our algorithm relies on a straightforward and scalable protocol for preparing the random state [25], which allows for circuit depths to be tuned to find a balance between desired accuracy and feasibility of execution on NISQ hardware. Furthermore, the algorithm is agnostic to implementation of the non-unitary transformation of the random state, which can be tailored to the resource constraints of different quantum devices.…”