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Real- and imaginary-time evolution with compressed quantum circuits

Sheng-Hsuan Lin, Rohit Dilip, Andrew G. Green, Adam Smith, Frank Pollmann

arXiv:2008.10322v2quant-phcond-mat.mes-hallcond-mat.str-el

TL;DR

The paper addresses how to represent and simulate quantum many-body states whose entanglement makes classical methods costly. It develops compressed sequential quantum circuits with variational real- and imaginary-time evolution, finding linear parameter-to-reachable-time scaling versus exponential MPS growth and demonstrating real-time dynamics on a QPU.

  • Problem

    Rapid entanglement growth limits classical MPS simulations of generic far-from-equilibrium quantum dynamics, motivating efficient near-term quantum representations.

  • Method

    The paper uses sequential two-qubit circuit ansätze motivated by MPS representations and variationally optimizes their gates during real- and imaginary-time evolution.

  • Results

    The circuit requires parameters growing linearly with reachable time while MPS parameters grow exponentially, and QPU magnetization closely matches exact results.

  • Takeaways & Limitations

    Compressed circuits provide efficient representations of physically relevant high-entanglement states and support time-evolution procedures implementable on existing quantum computers.

Abstract

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The current generation of noisy intermediate scale quantum computers introduces new opportunities to study quantum many-body systems. In this paper, we show that quantum circuits can provide a dramatically more efficient representation than current classical numerics of the quantum states generated under non-equilibrium quantum dynamics. For quantum circuits, we perform both real- and imaginary-time evolution using an optimization algorithm that is feasible on near-term quantum computers. We benchmark the algorithms by finding the ground state and simulating a global quench of the transverse field Ising model with a longitudinal field on a classical computer. Furthermore, we implement (classically optimized) gates on a quantum processing unit and demonstrate that our algorithm effectively captures real time evolution.

I. INTRODUCTION

The paper targets quantum many-body ground states and far-from-equilibrium dynamics, where exact diagonalization and MPS methods face scaling limits. It introduces compressed quantum circuits as efficient representations for high-entanglement, low-complexity states and develops near-term-compatible evolution methods.

  • Motivation: Exact diagonalization requires exponentially many parameters, while MPS methods become limited when entanglement grows rapidly during generic dynamics.These limitations motivate alternative representations for non-equilibrium quantum states.
  • Motivation: NISQ devices store quantum states with resources growing linearly rather than exponentially with system size, despite noise limiting many quantum algorithms.This makes near-term quantum simulation algorithms particularly relevant.
  • Motivation: Quantum circuits can represent states with high entanglement but low complexity, defining a complexity window beyond current classical numerical methods.The paper studies whether this distinction can support physically relevant quantum states.
  • Contribution: The paper studies circuits motivated by MPS representations, performs real- and imaginary-time evolution, and implements classically optimized gates on a QPU.The circuits require exponentially fewer parameters for a given amount of entanglement.
  • Approach: An order-M variational ansatz uses M gate layers and optimizes each gate to represent the time-evolved state within its circuit sub-manifold.The ansatz is described as capturing low-complexity, high-entanglement states.

II. COMPRESSED CIRCUITS

The compressed-circuit ansatz uses sequential two-qubit gates to capture the simple entanglement structure generated by local time evolution. It forms a restricted MPS sub-manifold with far fewer parameters and is tested on global quenches of a longitudinal-field Ising chain.

  • Ansatz structure: Sequential circuits contain two-qubit gates arranged in M layers, with total depth 2(M −1) + N −1 scaling linearly in system size N and order M.Unlike brickwall circuits, the ansatz does not restrict the correlation length of represented states.
  • Ansatz structure: The circuit states form an MPS sub-manifold with bond dimension χ = 2^M; M = 1 is exactly equivalent to bond dimension χ = 2.For M > 1, the circuit remains a restricted subset of the corresponding MPS manifold.
  • Parameter efficiency: For M > 1, these circuits use exponentially fewer parameters than a generic canonical MPS with bond dimension 2^M.They therefore describe high-entanglement, low-complexity states, although the reduction does not necessarily yield a sparse classical MPS representation.
  • Benchmark: Fast ballistic entanglement growth in far-from-equilibrium dynamics typically makes mid-to-long-time simulation difficult beyond small systems.This provides the stress test for the compressed-circuit representation.
  • Benchmark: The benchmark studies a global quench in a quantum Ising chain with transverse and longitudinal fields from a polarized initial state.The target is the real- or imaginary-time state |Ψ(t)⟩ = e^−iHt|Ψ⟩ after the quench.

A. Efficient Representation of Quantum States

Classical benchmarks compare optimized compressed circuits with high-bond-dimension MPS states generated by TEBD. The circuit approximation improves with order and shows linear parameter growth with reachable time, contrasting with exponential MPS growth.

  • Benchmark setup: The benchmark uses fourth-order Trotterized TEBD for N = 31, maximum bond dimension χ = 1024, and step size τ = 0.01 as a quasi-exact reference.Selected MPS states are compressed into optimized order-M quantum circuits.
  • Fidelity: Circuit fidelity decreases with time as correlations build, improves with increasing order M, and is higher for weaker longitudinal field h.The h = 0.9045 parameters are chosen for dynamics expected to be chaotic and difficult to simulate due to fast scrambling.
  • Entanglement: The ansatz captures rapid ballistic entanglement growth for small h, while entanglement growth slows as h increases.The paper associates larger h with increased practical state complexity despite slower entanglement growth.
  • Scaling: For fidelity F > 1 −10^−4, circuit parameters scale linearly with reachable time t*, whereas MPS parameters grow exponentially.The comparison uses t* defined by the fidelity threshold and includes multiple longitudinal-field values.
  • Scaling: Compressed circuits generically outperform fully Trotterized time evolution, with the quantitative improvement depending on model parameters.The reduced circuit depth is particularly valuable for NISQ devices where Trotterized evolution is challenging.

III. VARIATIONAL TIME EVOLUTION ALGORITHM

The variational time-evolution algorithm advances a circuit state by applying a Trotterized step and optimizing the circuit gates for maximum fidelity. The procedure is designed for hybrid execution and supports QPU demonstrations of real-time dynamics.

  • Quantum implementation: The method enables hybrid quantum optimization by optimizing gates classically and feeding them to a QPU to create the quantum state.The paper presents this as a route toward dynamics beyond classical state-storage limits.
  • Time-step construction: The algorithm applies a second-order Trotterized approximation to the evolution operator for each time step.The step is written as e^−iHeven∆t/2e^−iHodd∆te^−iHeven∆t/2.
  • Variational optimization: It iteratively optimizes the two-site gates defining the next circuit state, updating each gate with a polar-decomposition procedure.After multiple sweeps, the fidelity converges.

A. Real Time Evolution

The circuit-restricted real-time algorithm captures magnetization and entanglement growth for non-integrable quenches, with accuracy improving as circuit order increases. Its optimization uses Trotterized steps followed by projection onto a fixed-order circuit manifold.

  • Real-time performance: The algorithm captures magnetization for times that scale linearly with circuit order M.The evolution remains entirely within the fixed-order circuit ansatz.
  • Real-time performance: The circuits reproduce linear entanglement growth, while the saturation value increases linearly with M.The corresponding MPS representation requires O(2^M) parameters.
  • Algorithm: The method first applies finite Trotterized time steps and then optimizes the state within the circuit ansatz.This makes the procedure closer to tDMRG or TEBD than stochastic TDVP evolution.
  • Benchmark: Figure 4 tracks central-site magnetization and half-chain von Neumann entropy for N = 11, g = 1.4, and h = 0.1 across different orders M.The quench is non-integrable and exhibits fast linear entanglement growth.
  • Limitation: Efficient optimization is not guaranteed because large restricted-manifold variational problems can develop barren plateaux.The authors identify this as requiring further work for this class of algorithms.

B. Imaginary Time Evolution

The circuit ansatz is extended to imaginary-time evolution by compressing non-unitary Trotterized evolution into the circuit manifold. Classical benchmarks show convergence to the best energy attainable by the ansatz, with implementation designed to keep measurement costs linear in system size.

  • Method: Imaginary-time evolution applies e^(-H∆τ) through sequential compression back onto the circuit ansatz.The non-unitary evolution operator is approximated by products of two-qubit non-unitary gates using Trotterization.
  • Method: A unitary acting on one extra ancilla embeds each required non-unitary gate for quantum-device implementation.The embedding uses post-selection.
  • Method: A block C and scaling factor s are chosen so the first 2^N columns are mutually orthonormal, after which QR decomposition completes the unitary.This construction guarantees unitarity of the embedded operator.
  • Method: Applying and optimizing each two-qubit Trotterized gate separately makes the total measurements per time step scale linearly with system size.Implementing a full Trotter step at every optimization step would require linearly many ancillas and incur exponential post-selection cost.
  • Benchmark: The approach uses iterative energy minimization similar to VQE, with classical unitary updates obtained through polar decomposition.The benchmark treats the minimized ansatz energy as the best achievable performance.
  • Results: The imaginary-time algorithm successfully converges to the optimal energy attainable with the ansatz for circuit orders M = 1, 2, 3.The M = 1 results match DMRG, while χ = 4 MPS performs better for M = 2, 3 but the circuit errors remain below current NISQ thresholds.

C. Simulation on QPU

The authors demonstrate the compressed real-time circuit on a five-qubit IBM QPU by classically optimizing its gates and measuring the resulting states. The measured magnetization closely matches exact diagonalization, including long-time central-spin dynamics inaccessible to naive Trotterization.

  • Implementation: The QPU demonstration uses classically optimized gates to construct and measure the corresponding compressed state.The device is a five-qubit IBM-Q processor codenamed Bogota.
  • Benchmark: The benchmark initializes a five-qubit product state containing a single x-basis domain wall and quenches with g = 0.25 and h = 0.2.These parameters produce dynamics dominated by a single mobile domain wall and are well approximated by an order M = 1 circuit.
  • Results: The full-system x-magnetization measured on the QPU reproduces the periodic spreading and reconstitution of the domain wall seen in exact diagonalization.Figure 6(a) compares exact-diagonalization simulation with QPU measurements.
  • Results: The central-spin magnetization quantitatively matches exact diagonalization at long times.Figure 6(b) reports the central-qubit comparison, averaged over ten circuit realizations.
  • Significance: The demonstrated long-time access exceeds naive Trotterized evolution on the device, which would require circuit depth O(t).The QPU states are prepared from classically optimized gate sets.

IV. DISCUSSION

The paper presents sequential quantum circuits as efficient representations for physically relevant ground states and non-equilibrium dynamics, with time-evolution algorithms suitable for near-term quantum computers. It also identifies applications to unitary compression, quantum-complexity studies, and higher-dimensional dynamics.

  • Sequential quantum circuits efficiently represent ground states and non-equilibrium quantum states within a sparse corner of the larger MPS manifold.The ansatz can represent relevant states more efficiently than fixed-bond-dimension MPS representations for some physical states.
  • Time scales reachable by the circuit representation scale linearly with the number of circuit parameters, providing an exponential advantage over existing classical methods.
  • The time-evolution algorithm is implementable natively on existing quantum computers and supports both real- and imaginary-time evolution.Imaginary-time evolution can be used to obtain ground states on a quantum computer.
  • Fidelity maximization with polar decomposition can also approximate multi-qubit unitaries using sequences of 2-qubit gates and compress deep quantum circuits.These applications are presented as particularly relevant for current NISQ devices.
  • The procedure offers a practical tool for probing quantum complexity, including complexity windows and difficult-to-analyze state classes such as quantum scar and many-body localized states.The paper reports that the complexity window appears to shrink for non-integrable systems.
  • A direct generalization is to short-time dynamics in higher-dimensional systems, where classical numerics are generally difficult and larger-system behavior may become tractable on quantum computers.The algorithms are agnostic to the specific ansatz, motivating comparisons among different entanglement patterns.

Appendix A: Matrix-product states as quantum circuits

The appendix establishes an exact mapping from right-canonical MPS representations to sequential quantum circuits. The mapping relates MPS bond dimension to circuit gate size, after which larger unitaries can be approximated using 2-qubit gates.

  • Exact MPS-to-circuit mapping: An MPS of bond dimension χ maps exactly to a sequential circuit with (n + 1)-site unitaries, where n = log2 χ.For spin-1/2 systems, these are qubit circuits.
  • Exact MPS-to-circuit mapping: Right-canonical MPS tensors are isometries mapping |αk−1⟩ to |αk, ik⟩, and each isometry can be extended to a unitary acting on a normalized state |0k⟩.
  • Exact MPS-to-circuit mapping: The resulting unitaries act sequentially on an initial product state, while internally contracted virtual indices preserve the final state's dimension.
  • Gate size and bond dimension: For uniform bond dimension χ = 2^n, the corresponding unitaries act on log2 χ + 1 sites, linking virtual-index rank to circuit gate size.
  • Ansatz scope: The order-1 ansatz exactly corresponds to bond dimension χ = 2, whereas a two-layer brickwall circuit with the same number of 2-site gates has finite correlation length.
  • Gate decomposition: An MPS tensor can be exactly represented as a multi-qubit unitary and then approximated by a sequence of 2-qubit gates.
  • Gate decomposition: An order-M circuit provides a sparse representation of MPS with bond dimension 2^M by replacing an (M + 1)-site unitary with 2-site unitaries.

Appendix B: Classical simulation algorithm for quantum circuit

The appendix presents fidelity-based optimization and time-evolution algorithms for quantum-circuit states, alongside a construction for embedding non-unitary operators into larger unitaries.

  • Optimization and time evolution: The first algorithm maximizes fidelity between a target state and a state parameterized by circuit unitaries, while the second evolves states within the circuit ansatz.The time-evolution procedure repeatedly applies a Trotterized step and reoptimizes the circuit representation.
  • Optimization and time evolution: Each circuit gate is updated by iteratively solving its local optimization problem while all remaining gates stay fixed.The environment matrix is formed by excluding the pertinent unitary, and polar decomposition supplies the optimal update.
  • Optimization and time evolution: The time-evolution algorithm maximizes F = |⟨Ψ(t + ∆t)| V(∆t)|Ψ(t)⟩|^2 and advances iteratively from an initial state.At each step, it applies time-evolution gates, finds updated circuit gates, and accumulates the fidelity.
  • Error estimation: Two errors limit the simulation: Trotterization error can be reduced with smaller ∆t or higher-order decompositions, whereas projection error depends on the order-M ansatz.Projection error limits the time reachable within a specified error threshold.
  • Error estimation: The monitored total-error estimate is reliable when Trotterization error remains small and each step has fidelity close to 1; in Fig. 10, its threshold crossing matches observable deviation.The comparison uses the time when ⟨σz⟩ begins to deviate.
  • Non-unitary operator embedding: An arbitrary non-unitary N-qubit operator can be embedded in an (N + 1)-qubit unitary using an ancilla initialized and post-selected in |0⟩.The construction chooses a block C and scaling factor s to orthonormalize the first 2^N columns, then completes the unitary with QR decomposition.

Appendix D: Detailed data for parameter counting

The appendix details parameter counting for MPS and order-M circuit ansätze, confirming that the parameter correspondence depends on circuit depth and initialization redundancies.

  • Parameter counting: The appendix includes parameter counts needed to achieve fixed fidelity as a function of time for both MPS and quantum-circuit representations.The data are presented for MPS in Fig. 11(a) and quantum circuits in Fig. 11(b).
  • Parameter counting: The parameter count for a complex isometric matrix W ∈ C^n×p with n ≥ p is 2np − p^2 real independent parameters.The isometric condition W†W = 1 imposes p^2 independent real constraints.
  • Parameter counting: Circuit parameter counting accounts for redundant degrees of freedom caused by starting from the fixed state |000...00⟩.The first gate has 2d^2 − 1 = 7 parameters, while other first-layer gates have 2d^3 − d^2 = 12.
  • Parameter counting: A bond dimension χ = 2 MPS has the same parameter count as the order-M = 1 ansatz, while a two-layer brickwall circuit has fewer parameters.This confirms the parameter-reduction result stated in Appendix A.

Appendix E: Randomized circuits for QPU measurement

The appendix addresses QPU measurement fluctuations from decomposing two-site gates into finite universal gate sets and uses circuit gauge freedom to mitigate them.

  • QPU measurement: Two-site gates are decomposed into finitely many gates from a universal gate set when implemented on a QPU.Small perturbations in the original two-site gate can produce large changes in its decomposition.
  • QPU measurement: Figure 11 fits MPS parameter-count data with f(Jt*) = ae^{bJt*} + c and quantum-circuit data with f(Jt*) = a × (Jt*) + b.The fits are reported separately for h = 0, h = 0.1, and h = 0.9045.
  • QPU measurement: These decomposition differences cause large fluctuations in measured observables because of QPU imperfections.The effect arises from the sensitivity of the decomposition to small gate perturbations.
  • QPU measurement: The proposed mitigation averages over circuit gauge freedom by inserting identities represented by random unitaries and their complex conjugates.The averaging preserves the represented quantum states while addressing implementation sensitivity.
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