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Solving strongly correlated electron models on a quantum computer

Dave Wecker, Matthew B. Hastings, Nathan Wiebe, Bryan K. Clark, Chetan Nayak, Matthias Troyer

arXiv:1506.05135v2quant-phcond-mat.str-el

TL;DR

Determining phase diagrams and ground-state properties on quantum computers is harder than merely evolving a known state, especially for strongly correlated models. Using the Hubbard model, the paper presents a complete workflow for state preparation, adiabatic evolution, and measurement, with O(N) gates and O(log N) depth per evolution step and improved measurement scaling.

  • Problem

    Quantum simulation can evolve known states efficiently, but determining ground-state phase diagrams and experimentally relevant properties of strongly correlated systems remains challenging.

  • Method

    The paper prepares candidate broken-symmetry mean-field states and arbitrary Slater determinants, adiabatically evolves them to Hubbard-model ground states, and measures observables and correlation functions.

  • Results

    The complete workflow uses O(N) gates and O(log N) parallel depth per time step, while non-destructive measurements reduce runtime scaling from O(ϵ^-2) to O(ϵ^-1).

  • Takeaways & Limitations

    The strategy makes large Hubbard-model simulations feasible on quantum computers, with estimates of a few thousand qubits and preparation within a few seconds for N ≈1000 sites.

Abstract

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One of the main applications of future quantum computers will be the simulation of quantum models. While the evolution of a quantum state under a Hamiltonian is straightforward (if sometimes expensive), using quantum computers to determine the ground state phase diagram of a quantum model and the properties of its phases is more involved. Using the Hubbard model as a prototypical example, we here show all the steps necessary to determine its phase diagram and ground state properties on a quantum computer. In particular, we discuss strategies for efficiently determining and preparing the ground state of the Hubbard model starting from various mean-field states with broken symmetry. We present an efficient procedure to prepare arbitrary Slater determinants as initial states and present the complete set of quantum circuits needed to evolve from these to the ground state of the Hubbard model. We show that, using efficient nesting of the various terms each time step in the evolution can be performed with just $\mathcal{O}(N)$ gates and $\mathcal{O}(\log N)$ circuit depth. We give explicit circuits to measure arbitrary local observables and static and dynamic correlation functions, both in the time and frequency domain. We further present efficient non-destructive approaches to measurement that avoid the need to re-prepare the ground state after each measurement and that quadratically reduce the measurement error.

I. INTRODUCTION

The paper develops a quantum-computing strategy for determining the Hubbard model’s phase diagram and ground-state properties, addressing challenges beyond straightforward state evolution. It combines efficient state preparation, time evolution, and measurement procedures.

  • Motivation: The Hubbard model reduces electronic-structure complexity by retaining one orbital per unit cell and only local interactions and nearest-neighbor hopping.For a 20×20 lattice, the number of terms falls from about ∼10^12 to ∼10^3.
  • Scientific goal: Determining the phase diagram requires ground states across model parameters and quantities such as quasiparticle gaps and phase stiffness.A central target is identifying whether a superconducting phase exists.
  • State preparation: The approach starts from candidate broken-symmetry mean-field states, prepares them efficiently, and adiabatically evolves them toward the Hubbard-model ground state.Gap closing signals that a different initial state and annealing path should be tried.
  • Quantum evolution: Each time-evolution step requires O(N) gates and O(log N) parallel circuit depth through efficient decomposition and nesting of Hamiltonian terms.The paper supplies explicit circuits for implementing the evolution.
  • Measurement: The paper gives circuits for local observables and static and dynamic correlation functions, including frequency-domain importance sampling for dynamical measurements.The frequency-domain method complements the usual time-domain approach.
  • Measurement: Non-destructive measurements reduce repeated state preparation, with runtime scaling O(ϵ^-1) rather than O(ϵ^-2) for the naive approach.A relatively cheap phase-estimation procedure recreates the ground state after measurement.

B. The physics of the Hubbard model

The Hubbard model captures key regimes of strongly correlated electrons, including insulating and antiferromagnetic behavior, while motivating a quantum procedure to test candidate phases. The paper uses known initial states and adiabatic evolution to map the phase diagram.

  • Half-filling: At half-filling, the Hubbard model is metallic for U = 0 but insulating for every U > 0.Its small- and large-U insulating regimes have different energy scales and mechanisms.
  • Half-filling: For U ≪ t, antiferromagnetic order doubles the unit cell, while the antiferromagnetic moment and charge gap are exponentially small.The passage gives N ∼ Δcharge ∼ t e^-ct/U.
  • Half-filling: For U ≫ t, the charge gap scales as ∼U and virtual hopping produces antiferromagnetic correlations described at low energies by an effective Heisenberg model.The exchange coupling is J = 4t^2/U, with antiferromagnetic correlations appearing at the lower scale ∼J.
  • Doping: Away from half-filling, antiferromagnetic order is suppressed and eventually disappears, while the clean Hubbard model has nonzero conductivity.Real-material disorder can preserve insulating behavior close to half-filling.
  • Doping: The central cuprate question is whether the Hubbard model has a d_x2−y2 superconducting ground state near U/t ∼8 and dopings 0.05 ≲ x ≲ 0.25.If not, longer-ranged interactions, hopping, interlayer coupling, or phonons may need consideration.
  • Adiabatic preparation: The proposed protocol begins with known gapped ground states, including BCS or other ordered mean-field states and disconnected 2×2 plaquettes.Arbitrary Slater determinants can be prepared deterministically before adiabatic evolution.
  • Adiabatic preparation: If the adiabatic gap remains open, the initial state is inferred to share the final ground-state phase; gap closing prompts another candidate state and path.Repeating this process across parameters maps the phase diagram.

A. Adiabatic Evolution from Mean-Field States

The paper tests candidate broken-symmetry ground states by adiabatically deforming tractable mean-field or plaquette states into weakly perturbed Hubbard models. It also gives an efficient Givens-rotation procedure for preparing arbitrary Slater determinants as starting states.

  • Superconducting mean-field paths: A superconducting test adiabatically connects a BCS mean-field Hamiltonian to a weakly perturbed Hubbard model.The perturbations gap Goldstone modes and nodal fermions while remaining small enough not to bias superconducting tendencies.
  • Superconducting mean-field paths: A persistent pair field pins the superconducting phase, while a small d_xy term gaps nodal fermions during adiabatic evolution.The pair-field strength is controlled by w; the nodal gap is controlled by u and is chosen much smaller than xJ.
  • Alternative ordered states: Antiferromagnetic paths replace superconducting symmetry-breaking terms with a weak staggered magnetic field and diagnose phases through gap closing.The gap remains open for an antiferromagnetic ground state but becomes O(1/N^z/2) near a transition to superconductivity, up to finite-size effects.
  • Alternative ordered states: Comparing paths from distinct ordered states can reveal secondary long-ranged order coexisting with superconductivity.Such an order would force evolution from a purely superconducting initial state to encounter a phase transition.
  • Slater determinant preparation: Arbitrary Slater determinants are prepared by initializing a product state and applying at most nNe Givens rotations.Each Givens rotation is a simple two-qubit gate up to Jordan–Wigner strings, avoiding the inefficiency of Trotterizing the standard orbital-creation procedure.

3. Numerical Results

For an eight-site Hubbard model, the authors prepared the free-fermion Slater determinant with 14 Givens rotations and then annealed into the interacting model. Longer annealing increased phase-estimation success toward unity, while larger systems lack classically known exact ground-state energies for direct success checks.

  • Initial-state preparation: 14 Givens rotations prepared the eight-site free-fermion ground state for both spin species.The system consisted of two rows of four sites, with vertical couplings doubled to avoid an exact degeneracy.
  • Annealing results: Increasing annealing time raised the probability of projecting onto the ground state until it converged to 1.The anneal started at U = 0 with vertical hopping 2 and reduced that hopping to 1 while turning on interactions.
  • Annealing results: Annealing times for approximately 90% ground-state overlap were not significantly different from those for the plaquette-joining preparation approach.The comparison was made for this small Hubbard-model experiment.
  • Verification limitation: Exact diagonalization enabled success verification on the small system, but larger quantum simulations will not have access to the exact ground-state energy.The small instance was checked using classical Lanczos methods.

2. Preparing the ground states of four-site plaquettes

The paper prepares four-site plaquette ground states from simple product states through adiabatic schedules, then couples plaquettes while explicitly breaking reflection symmetry. It tests hole pairing by decoupling plaquettes and uses larger Trotter steps for preparation before higher-accuracy phase estimation.

  • Four-site plaquettes with two or four electrons are initialized in simple product states and adiabatically evolved into plaquette ground states.
  • The four-electron schedule starts with zero hopping and biased empty sites, ramps hoppings to t, then reduces the potentials to zero.The listed schedule uses stages T1 and T2.
  • T1 = T2 ≈10t^-1 and ϵ = U + 4t are sufficient for high-fidelity plaquette preparation.The authors note that phase estimation can project a moderately accurate annealed state into the ground state.
  • Coupling plaquettes: Coupling plaquettes requires an explicit symmetry-breaking potential because an asymmetric two-electron/four-electron initial state cannot adiabatically reach the symmetric ground state otherwise.The full lattice extends this strategy by biasing selected plaquettes before switching on inter-plaquette hoppings.
  • Decoupling plaquettes: For U = 2t and U = 4t, P33 decreases during decoupling, indicating hole pairing; for U = 6t and U = 8t, P33 increases, indicating no pairing.The observed crossover is consistent with the known four-site-plaquette critical value U ≈4.5t.
  • Trotter errors: 0.25t^-1 for annealing and smaller phase-estimation steps produced relative errors below 10^-3 at 0.05t^-1 and below 10^-5 at 0.003t^-1.The larger preparation step limited overlap to 80%–96% even for long annealing times.

IV. IMPLEMENTING UNITARY TIME EVOLUTION OF THE HUBBARD MODEL

The Hubbard-model evolution circuits implement repulsion, chemical-potential, hopping, and pairing terms after a Jordan–Wigner transformation. Ordering and nesting organize commuting terms for parallel execution and reduce circuit depth, including logarithmic-depth parity computation.

  • The circuits cover hopping terms, pairing terms, repulsion terms, and chemical-potential terms used during annealing.Pairing evolution is quadratic like hopping and differs mainly in basis-change gates.
  • The repulsion and chemical potential terms: Repulsion and chemical-potential terms commute within their respective sets, requiring O(N) gates and O(1) parallel circuit depth.
  • Pairing evolution: Pairing evolution for real Δ is implemented by mapping fermionic operators to Pauli X and Y operators together with Jordan–Wigner parity strings.The circuit may be controlled by an ancilla and uses basis changes analogous to hopping circuits.
  • Jordan–Wigner ordering: A snake ordering places all spin-up qubits before spin-down qubits and groups hopping terms into four sets, enabling parallel execution within each set.Horizontal terms need no Jordan–Wigner string, while vertical terms are nested.
  • Jordan–Wigner ordering: Tree-based parity computation reduces Jordan–Wigner-string depth to O(log(N)) for boundary terms, with total depth reducible to polylogarithmic in N.

V. FINDING THE GROUND STATE BY QUANTUM PHASE ESTIMATION

The paper combines adiabatic preparation with quantum phase estimation to project approximate states onto the Hubbard-model ground state. It also reduces phase-estimation cost and analyzes annealing success, improved paths, and Trotterized preparation costs.

  • Ground-state projection: Quantum phase estimation can project a state with reasonably large ground-state overlap, potentially outperforming slow annealing when the gap is small.The simplest annealing path scales inversely with the gap squared, whereas phase estimation scales inversely with the gap up to logarithmic corrections.
  • Reducing phase-estimation cost: A symmetric controlled-evolution modification gives a slightly larger than twofold depth reduction and a fourfold reduction in arbitrary rotations.The ancilla selects evolution by exp(iHt/2) or exp(−iHt/2), preserving the phase-estimation result while halving the evolution time.
  • Annealing-time estimates: For 1002 sites and annealing time t = 1000, the ground-state probability is 0.54, while for 2002 sites it falls to 0.13 at the same time.The lowest-energy mode dominates the small success probability, so small changes in the low-energy density of states can have dramatic effects.
  • Annealing-time estimates: About 10^4 Trotter steps and a parallel circuit depth of about one million gates suffice for the reported ground-state preparation estimate.Trotter time steps of order 0.25 were sufficient to obtain reasonable overlap, and each parallel step has depth growing logarithmically with system size.
  • Improved annealing paths: The cost of implementing the initial-state projection P0 scales as ϵ^(−1+o(1)) after choosing the preparation error δ proportional to ϵ^2/log(1/ϵ).The underlying preparation cost is sub-polynomial in 1/δ, while O(ϵ^−1 log(1/ϵ)) repetitions are required.
  • Trotterization: Numerical evidence suggests that high-order Trotter-Suzuki formulas are often unnecessary for the small Hubbard models considered.The paper notes that low-order methods may therefore be preferable in practice, while tight second-order error bounds are provided in the appendix.

VI. MEASURING OBSERVABLES

The measurement program targets densities, correlations, kinetic energies, Green’s functions, and pair correlations rather than only total energy. These observables characterize ground-state physics, including superconducting order and its relation to doping-dependent kinetic energy.

  • Measurement goals: The measurement section focuses on densities, density correlations, kinetic energies, Green’s functions, and pair correlation functions.Total energy can be measured by phase estimation but is described as the least interesting quantity for this purpose.
  • Physical interpretation: Pair-field correlations reveal superconducting properties through their long-distance behavior, while doping-dependent kinetic energy can test kinetic-energy gain as a possible superconductivity signature.The long-distance pair correlation determines the condensate fraction.

A. Local observables and equal-time correlations

Local densities and equal-time correlations can be extracted from computational-basis measurements, while unitary and stabilizer circuits provide broader correlation-measurement strategies. Commuting Pauli products enable nondestructive sequential measurements without recreating the state.

  • Local observables: Single-qubit measurements give local densities, two-qubit measurements give double occupancies, and four-qubit measurements give density and spin correlations.All qubits can be measured simultaneously in the computational basis to obtain these quantities from shared measurement data.
  • General unitary measurements: One-bit phase estimation measures the expectation value of any unitary with an available controlled implementation, including Hamiltonian terms.For hopping-related operators, θ = π distinguishes eigenvalue 0 from ±1, while θ = π/2 distinguishes +1 from −1.
  • Stabilizer strategy: The stabilizer strategy measures sums of mutually commuting Pauli products by replacing controlled Z-basis rotations with Z-basis measurements.Because the Pauli products commute, they can be measured in any order without recreating the state.
  • Stabilizer strategy: For the Hpq circuit, summing two ±1 Z-measurements and dividing by two estimates the hopping-operator expectation value.The division by two arises because the controlled unitaries rotate by half the coupling strength; pairing measurements use the difference instead.

4. The FSWAP Strategy for the kinetic energy and Green’s functions

The FSWAP strategy brings fermionic modes together so kinetic-energy, Green’s-function, and pair-correlation measurements can be performed with local circuits. Measurements can be parallelized across disjoint pairs and selected long-range pair separations.

  • Green’s functions: FSWAP gates move nonadjacent fermionic modes together while preserving the fermionic sign structure required for Green’s-function measurements.The qubits are swapped until the relevant modes are adjacent in the normal ordering, then a two-site measurement circuit is applied.
  • Kinetic energy: A basis change followed by Z measurements measures the kinetic energy on an adjacent bond.The bond kinetic energy is represented after the basis change by the corresponding hopping coefficient.
  • Parallel measurements: With N qubits, N/2 disjoint measurements can be performed simultaneously.No qubit may participate in two measurements in the same parallel layer.
  • Pair correlations: Four-point pair correlations are measured by FSWAPing four modes into adjacent positions and applying two Green’s-function measurement circuits.The resulting estimator uses the measured Z components together with indicators for the relevant particle-occupation sectors.
  • Pair correlations: Long-range pair order can be tested without measuring every N^4 four-point function by using nearby sites within each pair and maximizing separation between pairs.This selection reflects the expectation that Hubbard-model pairs are tightly bound.

B. Dynamic correlation functions and gaps

The paper gives circuits for measuring dynamic correlation functions in time and frequency domains, including local, momentum-resolved, and spectral observables.

  • Time-domain measurements: Unitary operator products can be measured with controlled operations and ancilla readout in the X basis.The circuit can avoid controlling time evolution, and one final evolution can be replaced by an ancilla Z rotation.
  • Frequency-domain measurements: Phase estimation on A|ψ0⟩ directly samples excited-state energies with weights |⟨ψn|A|ψ0⟩|2, producing a histogram of the dynamic structure factor.This importance-sampling approach avoids resolving delta functions by sampling the correlation function at many times and Fourier transforming.
  • Local correlations: Local dynamical spin-density correlations are obtained by choosing A = Zp,σ and relating spin-density operators to particle-number correlations.Time-independent terms contribute only at zero frequency, while local charge and spin correlations follow from the same construction.
  • Spectral functions: The same circuits measure arbitrary single-particle wave functions, including momentum eigenstates for momentum-resolved electron and hole spectral functions.Particle versus hole excitations can be identified by measuring the total particle number after phase estimation.

VII. NON-DESTRUCTIVE MEASUREMENTS

The paper develops non-destructive measurement strategies for ground-state observables, while noting that adiabatic preparation can become the dominant cost for small spectral gaps.

  • Preparation cost: For small simulated systems, adiabatic preparation is faster than phase estimation, but linear annealing is expected to scale as Δ^-2 when the spectral gap is Δ.Energy resolution by phase estimation scales as 1/Δ, so preparation can dominate as system size increases; higher-order paths alleviate this problem.
  • Measurement strategies: Non-destructive measurement uses phase estimation to recover the ground state after measurements, avoiding repeated preparation from scratch.A Hellmann-Feynman approach and a recovery-based approach are presented for different circumstances.
  • Hellmann-Feynman approach: Adiabatically adding λO to H and measuring the perturbed ground-state energy yields the expectation value of O without destroying the original ground state.For a Green’s function, the perturbation is implemented as a hopping term λ(c†q,σcp,σ).
  • Error scaling: O(ϵ^-2) phase-estimation time gives only a constant improvement over repeated destructive measurements for the basic estimator.A symmetric derivative permits λ = O(ϵ^1/2) and time O(ϵ^-3/2), while higher-order estimators approach a near-quadratic speedup.

B. Non-Destructive Measurements With Quadratic Speedup

The paper analyzes recovery-based non-destructive measurements using projective measurements and phase estimation, then derives a quadratic improvement in sampling effort.

  • Quadratic speedup: The improved protocol uses phase estimation to accelerate non-destructive measurement quadratically, up to logarithmic factors.The initial binary-sampling approach requires a number of samples proportional to the square of the inverse error.
  • Recovery Map: The recovery protocol measures Q, projects onto the ground state with P0, and repeats until the ground state is restored.After an outcome i, the post-measurement state is φi, and the restoration probability is |ai|2.
  • Recovery Map: The repeated recovery process is represented as a Markov chain over measurement outcomes, with transitions determined by the probabilities of Q and P0.The analysis tracks transitions that return directly to the ground state and those that continue to another outcome.
  • Recovery Map: For k possible outcomes, the expected number of steps before a transition is k/2, and half of transitions return to the ground state.The derivation uses a probability reinterpretation valid initially for |ai|2 ≤ 1/2 and extends the result analytically.
  • Quadratic speedup: Both the required phase estimations and measurements of Q equal k in the analyzed protocol.

2. Quadratic Speedup

The paper develops non-destructive measurement strategies that preserve or recover the ground state, enabling repeated observable measurements without re-preparing it each time.

  • Quadratic Speedup: The measurement procedure requires a rank-one ground-state projector P0 and alternates measurements of Q and P0 to recover the ground state.The analysis is performed in the two-dimensional space spanned by ψ0 and Qψ0.
  • Quadratic Speedup: Phase estimation of a controlled product of reflections yields the ground-state projector expectation from either eigenvalue outcome.The two eigenvalues differ only in sign, but both return the same value for ⟨ψ0|Q|ψ0⟩.
  • Quadratic Speedup: Coherent phase estimation implements P0 by storing the energy estimate coherently and flagging whether it lies near the known ground-state energy.The control-bit outcomes remain unmeasured during this procedure.
  • Quadratic Speedup: The additional qubit overhead for measuring P0 scales logarithmically when the required energy accuracy is polynomially small in system size.More accurate implementation of a projective measurement also incurs only logarithmic control-bit overhead.
  • Quadratic Speedup: An alternative constructs an approximate ground-state projector by transporting an easily implemented free-fermion projector along an adiabatic path.The accuracy depends on the annealing path, and improved paths can provide a near-quadratic speedup.
  • Quadratic Speedup: The broader strategy combines trial-state preparation, adiabatic evolution, phase estimation, parallelized time evolution, and non-destructive measurements for Hubbard-model studies.The paper also proposes frequency-space measurements of dynamic structure factors and spectral functions.

Appendix A: Error bounds for time-dependent Trotter Formulas

Appendix A analyzes Trotter errors for time-dependent Hamiltonians, emphasizing that adiabatic evolution can make piecewise-constant approximations non-negligible and deriving bounds for the second-order formula.

  • Appendix A: Error bounds for time-dependent Trotter Formulas: Time-dependent Hamiltonian simulation is subtler than time-independent simulation, particularly for adiabatic state preparation.Results established for time-independent dynamics do not necessarily transfer to the time-dependent case.
  • Appendix A: Error bounds for time-dependent Trotter Formulas: Tighter error-scaling estimates may be important because the relevant fermionic Hamiltonians contain many terms.This affects estimates of the cost of adiabatic state preparation.
  • Appendix A: Error bounds for time-dependent Trotter Formulas: Trotter errors from piecewise-constant Hamiltonians cannot generally be neglected merely because an adiabatic evolution is slow.Adiabatic conditions constrain derivatives relative to a power of the minimum eigenvalue gap, and high-accuracy preparation may expose these errors.
  • Appendix A: Error bounds for time-dependent Trotter Formulas: The appendix generalizes second-order Trotter-Suzuki error bounds from time-independent to time-dependent Hamiltonians.The evolution from time 0 to 1 is approximated by a single second-order Trotter-Suzuki step in the analyzed bound.
  • Appendix A: Error bounds for time-dependent Trotter Formulas: The derived time-dependent bound reduces to the known time-independent bound and matches its asymptotic error scaling up to small factors.The comparison uses the triangle inequality and a small multiplicative factor in the time-independent limit.
  • Appendix A: Error bounds for time-dependent Trotter Formulas: The proof bounds the Trotter approximation in two stages and corrects a prior typographical error involving a sum of distinct operator norms.The correction does not change the scaling results cited from later work.
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