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arXiv:2312.13892 · Quantum

Efficient Quantum Algorithm for Filtering Product States

Reinis Irmejs, Mari Carmen Bañuls, J. I. Cirac

Published Jun 27, 2024• 9 citations
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Abstract

We introduce a quantum algorithm to efficiently prepare states with a small energy variance at the target energy. We achieve it by filtering a product state at the given energy with a Lorentzian filter of width δ . Given a local Hamiltonian on N qubits, we construct a parent Hamiltonian whose ground state corresponds to the filtered product state with variable energy variance proportional to δN . We prove that the parent Hamiltonian is gapped and its ground state can be efficiently implemented in poly(N,1/δ) time via adiabatic evolution. We numerically benchmark the algorithm for a particular non-integrable model and find that the adiabatic evolution time to prepare the filtered state with a width δ is independent of the system size N . Furthermore, the adiabatic evolution can be implemented with circuit depth O(N2δ−4) . Our algorithm provides a way to study the finite energy regime of many body systems in quantum simulators by directly preparing a finite energy state, providing access to an approximation of the microcanonical properties at an arbitrary energy.

DOI
10.22331/q-2024-06-27-1389

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We introduce a quantum algorithm to efficiently prepare states with a small energy variance at the target energy. We achieve it by filtering a product state at the given energy with a Lorentzian filter of width δ . Given a local Hamiltonian on N qubits, we construct a parent Hamiltonian whose ground state corresponds to the filtered product state with variable energy variance proportional to δN . We prove that the parent Hamiltonian is gapped and its ground state can be efficiently implemented in poly(N,1/δ) time via adiabatic evolution. We numerically benchmark the algorithm for a particular non-integrable model and find that the adiabatic evolution time to prepare the filtered state with a width δ is independent of the system size N . Furthermore, the adiabatic evolution can be implemented with circuit depth O(N2δ−4) . Our algorithm provides a way to study the finite energy regime of many body systems in quantum simulators by directly preparing a finite energy state, providing access to an approximation of the microcanonical properties at an arbitrary energy.
We propose applying the adiabatic algorithm to prepare high-energy eigenstates of integrable models on a quantum computer. We first review the standard adiabatic algorithm to prepare ground states in each magnetization sector of the prototypical XXZ Heisenberg chain. Based on the thermodynamic Bethe ansatz, we show that the algorithm circuit depth is polynomial in the number of qubits N , outperforming previous methods explicitly relying on integrability. Next, we propose a protocol to prepare arbitrary eigenstates of integrable models that satisfy certain conditions. For a given target eigenstate, we construct a suitable parent Hamiltonian written in terms of a complete set of local conserved quantities. We propose using such Hamiltonian as an input for an adiabatic algorithm. After benchmarking this construction in the case of the non-interacting XY spin chain, where we can rigorously prove its efficiency, we apply it to prepare arbitrary eigenstates of the Richardson-Gaudin models. In this case, we provide numerical evidence that the circuit depth of our algorithm is polynomial in N for all eigenstates, despite the models being interacting.
Eigenstate filters underpin near-optimal quantum algorithms for ground-state preparation. Their realization on current quantum computers, however, poses a challenge as the filters are typically represented by deep quantum circuits. Additionally, since the filters are created probabilistically, their circuits need to be rerun many times when the associated success probability is small. Here, we describe a strategy to implement a ground-state filter on quantum hardware in the presence of noise by prepending the filter with digitized adiabatic quantum computing. The adiabatically prepared input state increases the success probability of the filter and also reduces its circuit depth requirements. At the same time, the filter enhances the accuracy of the adiabatically prepared ground state. We compare the approach to the purely adiabatic protocol through numerical simulations and experiments on the Quantinuum H1-1 quantum computer. We demonstrate a significant improvement in ground-state accuracies for paradigmatic quantum spin models.
We devise a quantum-circuit algorithm to solve the ground state and ground energy of artificial graphene. The algorithm implements a Trotterized adiabatic evolution from a purely tight-binding Hamiltonian to one including kinetic, spin-orbit, and Coulomb terms. The initial state is obtained efficiently using Gaussian-state preparation, while the readout of the ground energy is organized into seventeen sets of measurements, irrespective of the size of the problem. The total depth of the corresponding quantum circuit scales polynomially with the size of the system. A full simulation of the algorithm is performed and ground energies are obtained for lattices with up to four hexagons. Our results are benchmarked with exact diagonalization for systems with one and two hexagons. For larger systems we use the exact state vector and approximate matrix product state simulation techniques. The latter allows us to systematically trade off precision with memory and therefore to tackle larger systems. We analyze adiabatic and Trotterization errors, providing estimates for optimal periods and time discretizations given a finite accuracy. In the case of large systems we also study approximation errors.
In this work we propose an approach for implementing time-evolution of a quantum system using product formulas. The quantum algorithms we develop have provably better scaling (in terms of gate complexity and circuit depth) than a naive application of well-known Trotter formulas, for systems where the evolution is determined by a Hamiltonian with different energy scales (i.e., one part is "large" and another part is "small"). Our algorithms generate a decomposition of the evolution operator into a product of simple unitaries that are directly implementable on a quantum computer. Although the theoretical scaling is suboptimal compared with state-of-the-art algorithms (e.g., quantum signal processing), the performance of the algorithms we propose is highly competitive in practice. We illustrate this via extensive numerical simulations for several models. For instance, in the strong-field regime of the 1D transverse-field Ising model, our algorithms achieve an improvement of one order of magnitude in both the system size and evolution time that can be simulated with a fixed budget of 1000 arbitrary 2-qubit gates, compared with standard Trotter formulas.
Quantum mechanical ground states of many-body systems can be important resources for various investigations: for quantum sensing, for benchmarking quantum hardware with classically solvable states, as the initial states for nonequilibrium quantum dynamics following quenches, the simulation of quantum processes that start by coupling systems in ground states, eg, could be a process in quantum chemistry, while their approximations are required as inputs to quantum phase estimation algorithm. However, preparing ground states can be challenging; for example, it may require adiabatic switching of Hamiltonian terms slower than an inverse gap, which can be time consuming and bring in decoherence. Here we investigate the possibility of preparing a many-body entangled ground state of a certain Hamiltonian, which can be called a quantum ``problem'' Hamiltonian, using the time evolution of an initial fiducial state by another time independent ``solver'' Hamiltonian with couplings up to unit strength for a very short fixed (unit) time: a ``time efficient'' ansatz. The parameters of the solver Hamiltonian are optimised classically minimising energy as the cost function. We present a study of up to n=14n=14n=14 qubit many-body states prepared using this methodology. Importantly, we find that a strategy of combining a warm start (an already prepared ground state of a n−1n-1n−1 qubit Hamiltonian) and incrementally adding extra couplings of a qubit is the best scaling strategy to prepare the ground state of a nnn-qubit Hamiltonian.
We propose a method to calculate finite-temperature properties of a quantum many-body system for a microcanonical ensemble by introducing a pure quantum state named here an energy-filtered random-phase state, which is also a potentially promising application of near-term quantum computers. In our formalism, a microcanonical ensemble is specified by two parameters, i.e., the energy of the system and its associated energy window. Accordingly, the density of states is expressed as a sum of Gaussians centered at the target energy with its spread corresponding to the width of the energy window. We then show that the thermodynamic quantities such as entropy and temperature are calculated by evaluating the trace of the time-evolution operator and the trace of the time-evolution operator multiplied by the Hamiltonian of the system. We also describe how these traces can be evaluated using random diagonal-unitary circuits appropriate to quantum computation. The pure quantum state representing our microcanonical ensemble is related to a state of the form introduced by Wall and Neuhauser for the filter diagonalization method [M. R. Wall and D. Neuhauser, J. Chem. Phys. 102, 8011 (1995)], and therefore we refer to it as an energy-filtered random-phase state. The energy-filtered random-phase state is essentially a Fourier transform of a time-evolved state whose initial state is prepared as a random-phase state, and the cut-off time in the time-integral for the Fourier transform sets the inverse of the width of the energy window. The proposed method is demonstrated numerically by calculating thermodynamic quantities for the one-dimensional spin-1/2 Heisenberg model on small clusters up to 28 qubits, showing that the method is most efficient for the target energy around which the dense distribution of energy eigenstates is found.