We present a holographic quantum simulation algorithm to variationally prepare thermal states of $d$-dimensional interacting quantum many-body systems, using only enough hardware qubits to represent a $(d\ensuremath{-}1)$-dimensional cross section. This technique implements the thermal state by approximately unraveling the quantum matrix-product density operator (qMPDO) into a stochastic mixture of quantum matrix-product states (sto-qMPS). The parameters of the quantum circuits generating the qMPS and of the probability distribution generating the stochastic mixture are determined through a variational optimization procedure. We demonstrate a small-scale proof-of-principle demonstration of this technique on Quantinuum's trapped-ion quantum processor to simulate thermal properties of correlated spin chains over a wide temperature range using only a single pair of hardware qubits. Then, through classical simulations, we explore the representational power of two versions of sto-qMPS ansatzes for larger and deeper circuits and establish empirical relationships between the circuit resources and the accuracy of the variational free energy.
We present a holographic quantum simulation algorithm to variationally prepare thermal states of d-dimensional interacting quantum many-body systems, using only enough hardware qubits to represent a (d\ensuremath−1)-dimensional cross section. This technique implements the thermal state by approximately unraveling the quantum matrix-product density operator (qMPDO) into a stochastic mixture of quantum matrix-product states (sto-qMPS). The parameters of the quantum circuits generating the qMPS and of the probability distribution generating the stochastic mixture are determined through a variational optimization procedure. We demonstrate a small-scale proof-of-principle demonstration of this technique on Quantinuum's trapped-ion quantum processor to simulate thermal properties of correlated spin chains over a wide temperature range using only a single pair of hardware qubits. Then, through classical simulations, we explore the representational power of two versions of sto-qMPS ansatzes for larger and deeper circuits and establish empirical relationships between the circuit resources and the accuracy of the variational free energy.
We present a suite of ``holographic'' quantum algorithms for efficient ground-state preparation and dynamical evolution of correlated spin systems, which require far fewer qubits than the number of spins being simulated. The algorithms exploit the equivalence between matrix-product states (MPS) and quantum channels, along with partial measurement and qubit reuse, in order to simulate a D-dimensional spin system using only a $(D\ensuremath{-}1)$-dimensional subset of qubits along with an ancillary qubit register whose size scales logarithmically in the amount of entanglement present in the simulated state. Ground states can either be directly prepared from a known MPS representation or obtained via a holographic variational quantum eigensolver (holoVQE). Dynamics of MPS under local Hamiltonians for time t can also be simulated with an additional (multiplicative) poly(t) overhead in qubit resources. These techniques open the door to efficient quantum simulation of MPS with exponentially large bond dimension, including ground states of two- and three-dimensional systems, or thermalizing dynamics with rapid entanglement growth. As a demonstration of the potential resource savings, we implement a holoVQE simulation of the antiferromagnetic Heisenberg chain on a trapped-ion quantum computer, achieving within 10(3) of the exact ground-state energy of an infinite chain using only a pair of qubits.
While standard approaches to quantum simulation require a number of qubits proportional to the number of simulated particles, current noisy quantum computers are limited to tens of qubits.With the technique of holographic quantum simulation, a D-dimensional system can be simulated with a (D -1)-dimensional subset of qubits, enabling the study of systems significantly larger than current quantum computers.Using circuits derived from the multiscale entanglement renormalization ansatz (MERA), we accurately prepare the ground state of an L = 32 critical, nonintegrable perturbed Ising model and measure long-range correlations on the ten-qubit Quantinuum trapped-ion computer.We introduce generalized MERA networks that interpolate between MERA and matrix product state networks and demonstrate that generalized MERA can capture far longer correlations than a MERA with the same number of qubits, at the expense of greater circuit depth.Finally, we perform noisy simulations of these two network ansatzes and find that the optimal choice of network depends on the noise level, available qubits, and the state to be represented.
Quantum computers promise to solve finite-temperature properties of quantum many-body systems, which is generally challenging for classical computers due to high computational complexities. Here, we report experimental preparations of Gibbs states and excited states of Heisenberg XX and XXZ models by using a 5-qubit programmable superconducting processor. In the experiments, we apply a hybrid quantum–classical algorithm to generate finite temperature states with classical probability models and variational quantum circuits. We reveal that the Hamiltonians can be fully diagonalized with optimized quantum circuits, which enable us to prepare excited states at arbitrary energy density. We demonstrate that the approach has a self-verifying feature and can estimate fundamental thermal observables with a small statistical error. Based on numerical results, we further show that the time complexity of our approach scales polynomially in the number of qubits, revealing its potential in solving large-scale problems.
Abstract Classical simulations of quantum circuits are vital for assessing potential quantum advantage and benchmarking devices, yet they require sophisticated methods to avoid the exponential growth of resources. Tensor network approaches, in particular matrix product states (MPS) combined with the time-evolving block decimation (TEBD) algorithm, currently dominate large-scale circuit simulations. These methods scale efficiently when entanglement is limited but suffer rapid bond dimension growth with increasing entanglement and handle long-range gates via costly SWAP insertions. Motivated by the success of the time-dependent variational principle (TDVP) in many-body physics, we reinterpret quantum circuits as series of discrete time evolutions, using gate generators to construct an MPS-based circuit simulation via a local TDVP formulation. This addresses TEBD's key limitations by (1) naturally accommodating long-range gates and (2) optimally representing states on the MPS manifold. By diffusing entanglement more globally, the method suppresses local bond growth and reduces memory and runtime costs. We benchmark the approach on five 49-qubit circuits: three Hamiltonian circuits (1D open and periodic Heisenberg, 2D 7x7 Ising) and two algorithmic ones (quantum approximate optimization, hardware-efficient ansatz). Across all cases, our method yields substantial resource reductions over standard tools, establishing a new state-of-the-art for circuit simulation and enabling advances across quantum computing, condensed matter, and beyond.
We demonstrate the use of matrix product state (MPS) models for discriminating quantum data on quantum computers using holographic algorithms, focusing on the problem of classifying a translationally invariant quantum state based on L qubits of quantum data extracted from it. We detail a process in which data from single-shot experimental measurements are used to optimize an isometric tensor network, the isometric tensors are compiled into unitary quantum operations using greedy compilation heuristics, parameter optimization on the resulting quantum circuit model removes the postselection requirements of the isometric tensor model, and the resulting quantum model is inferenced on either product state (single-shot measurement) or entangled quantum data. We demonstrate our training and inference architecture on a synthetic dataset of six-site single-shot measurements from the bulk of a one-dimensional transverse field Ising model (TFIM) deep in its antiferromagnetic and paramagnetic phases. We find that increasing the bond dimension of the tensor-network model, amounting to adding more ancilla qubits to the circuit representation, improves both the average number of correct classifications across the dataset and the single-shot probability of correct classification. We experimentally evaluate models on Quantinuum's H1-2 trapped ion quantum computer using entangled input data modeled as translationally invariant, bond dimension 4 MPSs across the known quantum phase transition of the TFIM. Using linear regression on the experimental data near the transition point, we find predictions for the critical transverse field of h=0.962 and 0.994 for tensor-network discriminators of bond dimension $\ensuremath{\chi}=2$ and $\ensuremath{\chi}=4$, respectively. These predictions compare favorably with the known transition location of h=1 despite training on data far from the transition point. Our techniques identify families of short-depth variational quantum circuits in a data-driven and hardware-aware fashion and robust classical techniques to precondition the model parameters, and can be adapted beyond machine learning to myriad applications of tensor networks on quantum computers, such as quantum simulation and error correction.
Tensor network methods, most prominently matrix product states (MPS), have become fundamental tools in modern quantum many-body physics. While MPS and extensions like the multiscale entanglement renormalization ansatz (MERA) and tree tensor networks (TTN) efficiently capture area-law entanglement and its logarithmic violations, they inherently struggle to represent highly entangled wavefunctions. Specifically, reaching the volume-law regime typically demands exponential resources within these conventional frameworks. Motivated by this challenge, we propose holographic isometric tensor network states (holographic isoTNS) that simulate quantum lattice models in D spatial dimensions via (D+1)-dimensional networks of tensors. The additional dimension substantially enlarges the representational manifold, while isometric constraints on each tensor ensure efficient contractibility. Using one-dimensional systems as testbeds, we analyze the properties of holographic isoTNS. First, we show that randomly initialized holographic isoTNS typically display volume-law entanglement at fixed bond dimension. Second, through analytic constructions and variational optimization, we demonstrate that holographic isoTNS can faithfully describe a broad class of highly entangled yet low-complexity states. In particular, the ansatz can represent arbitrary fermionic Gaussian states, Clifford states, extensions of rainbow states, and certain short-time-evolved states under local evolution. Third, to exploit this expressivity in broader contexts, we implement a time-evolving block decimation (TEBD) algorithm on holographic isoTNS. While the method remains efficient and scalable, error accumulation over TEBD sweeps suggests the need for further algorithmic improvement. Overall, holographic isoTNS broaden the scope of tensor-network methods, opening new avenues to study physics in the volume-law regime.