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Encoding of Matrix Product States into Quantum Circuits of One- and Two-Qubit Gates

Shi-Ju Ran

arXiv:1908.07958v2quant-phcond-mat.str-el

TL;DR

Large, highly entangled MPSs are difficult to realize on quantum hardware because their virtual dimensions can require high-level qudits or inefficient multi-qubit-gate compilation. The paper constructs unitary matrix product disentanglers that encode MPSs using only one- and two-qubit gates, achieving high-fidelity preparation with fewer qubits than the target MPS sizes. The method is benchmarked on strongly correlated spin-model ground-state MPSs, while classical simulations can become exponentially costly as circuit depth increases.

  • Problem

    Large, highly entangled MPSs are difficult to realize because their virtual dimensions can require high-level qudits or inefficient multi-qubit-gate compilation.

  • Method

    The paper constructs unitary matrix product disentanglers that form one- and two-qubit circuits evolving a product state into a target MPS.

  • Results

    High fidelity is reached with O(10) layers, and qubit-efficient circuits encode benchmark MPSs using fewer than 10 qubits; encoding errors further decrease with depth D.

  • Takeaways & Limitations

    The constructed circuits can accurately and efficiently prepare strongly entangled MPSs, including states with large virtual dimensions or system sizes.

Abstract

from arXiv · show

The matrix product state (MPS) belongs to the most important mathematical models in, for example, condensed matter physics and quantum information sciences. However, to realize an $N$-qubit MPS with large $N$ and large entanglement on a quantum platform is extremely challenging, since it requires high-level qudits or multi-body gates of two-level qubits to carry the entanglement. In this work, an efficient method that accurately encodes a given MPS into a quantum circuit with only one- and two-qubit gates is proposed. The idea is to construct the unitary matrix product operators that optimally disentangle the MPS to a product state. These matrix product operators form the quantum circuit that evolves a product state to the targeted MPS with a high fidelity. Our benchmark on the ground-state MPS's of the strongly-correlated spin models show that the constructed quantum circuits can encode the MPS's with much fewer qubits than the sizes of the MPS's themselves. This method paves a feasible and efficient path to realizing quantum many-body states and other MPS-based models as quantum circuits on the near-term quantum platforms.

I. INTRODUCTION

MPSs are powerful models for physics and quantum information, but realizing highly entangled, large MPSs on quantum hardware is difficult because their virtual dimensions can greatly exceed physical dimensions and naive compilation is inefficient. The paper proposes matrix product disentanglers that encode such MPSs into one- and two-qubit circuits with high fidelity and substantially fewer qubits.

  • Motivation: MPSs efficiently describe important states in condensed matter physics and quantum information, including ground, thermal, GHZ, and AKLT states.They are also used in statistical physics, non-equilibrium quantum physics, field theories, and machine learning.
  • Motivation: Realizing MPSs on quantum hardware is limited by short coherent times and small available qubit counts.The paper notes recent progress, including GHZ-state realization up to twenty qubits.
  • Hardware challenge: Virtual dimensions χ are generally much larger than physical dimensions d, making direct realization require impractical χ-level qudits, often with χ ∼O(10^2) or larger.The virtual degrees of freedom carry the MPS entanglement.
  • Hardware challenge: Replacing χ-level qudits with multiple-qubit gates still requires compilation, whose circuit depth generally scales polynomially with χ.This motivates efficient encoding algorithms using only one- and two-qubit gates.
  • Approach: The proposed algorithm constructs unitary matrix product operators, called matrix product disentanglers, that disentangle a target MPS and form a circuit evolving a product state into it.The method targets MPSs with d = 2 and χ ≫d using only one- and two-qubit gates.
  • Results: High fidelity is reached with only O(10) disentangler layers, while a qubit-efficient implementation encodes benchmark MPSs using fewer than 10 qubits.The benchmark uses MPSs approximating ground states of one-dimensional strongly correlated spin systems.

II. MATRIX PRODUCT STATE AND ORTHOGONAL FORM: PRELIMINARIES

An MPS is organized by physical and virtual indices, with orthogonal forms providing the normalization structure used to construct matrix product disentanglers. The resulting unitary circuit uses one- and two-qubit gates and can accurately approximate highly entangled MPSs.

  • MATRIX PRODUCT STATE AND ORTHOGONAL FORM: PRELIMINARIES: An MPS uses physical indices for local Hilbert spaces and virtual indices whose dimension χ controls the represented entanglement.The physical dimension is d, while virtual dimensions are bounded by χ and are generally chosen much larger than d.
  • MATRIX PRODUCT STATE AND ORTHOGONAL FORM: PRELIMINARIES: Any MPS can be transformed into left-orthogonal form by a gauge transformation, with orthogonal conditions fixing the tensor normalization structure.These conditions also determine renormalization-group flow directions in Hilbert space.
  • MATRIX PRODUCT STATE AND ORTHOGONAL FORM: PRELIMINARIES: For d = χ = 2, a left-orthogonal MPS can be exactly encoded by a single-layer circuit of one- and two-body gates on qudits, which are qubits when d = 2.The single-layer construction is introduced as the starting point for the deeper circuit scheme.
  • MATRIX PRODUCT STATE AND ORTHOGONAL FORM: PRELIMINARIES: For χ > d, truncating the MPS to χ = d yields a disentangler that reduces rather than fully removes the target state's entanglement.The procedure first selects the χ = d MPS maximizing fidelity with the original state, then constructs the unitary from that approximation.
  • MATRIX PRODUCT STATE AND ORTHOGONAL FORM: PRELIMINARIES: F1 is about O(10) ∼ O(10^4) times smaller than F0, indicating that the constructed circuit evolves |0⟩ close to the targeted MPS even when χ ≫ d and N ≫ 1.F0 measures distance from the best separable state, whereas F1 measures distance from the circuit-generated state.

III. ENCODING MATRIX PRODUCT STATE INTO DEEP QUANTUM CIRCUIT

The algorithm iteratively constructs matrix product disentanglers to encode MPS ground states into deep circuits, improving fidelity with circuit depth while compressing coefficients and qubit requirements.

  • Encoding algorithm: The algorithm repeatedly truncates each intermediate MPS to χ = d, computes an MPD, and applies it before returning the resulting D disentanglers.This iterative procedure increases the circuit depth until the preset layer count is reached.
  • Benchmarks: The encoding is tested on ground-state MPSs of one-dimensional transverse Ising, Heisenberg, and XY models, including gapless cases requiring large χ.The gapless models’ large entanglement makes them challenging targets for existing methods.
  • Fidelity and accuracy: FD decreases with circuit depth, with the largest reduction at D = 1 and an additional 20%–40% reduction at D = 9 relative to F1.The reported reductions are for the NLF between the MPS and the circuit-evolved state.
  • Fidelity and accuracy: For the critical transverse Ising model, FD initially increases with system size N and then converges for D = 0, 1, and 9.The inset compares the accuracy improvement from D = 1 to D = 9 through F9/F1.
  • Resource compression: For χ = 64, the coefficient count per site decreases from 2^13 for the original MPS to 2^7 for the circuit at D = 8.The number of gates and qubits scales linearly with N and D, while the qubit-efficient scheme makes qubit count independent of N and linear in D.
  • Resource compression: The NLF rises suddenly when D exceeds log2 ˜χ, while increasing the dimension cutoff shifts this threshold.The figure compares FD across circuit depth for different cutoffs ˜χ.

IV. ERROR PROPAGATIONS AND COMPLEXITY IN CLASSICAL SIMULATIONS

Classical simulation remains controlled only up to a cutoff-dependent depth because intermediate virtual dimensions grow exponentially and truncation errors can propagate sharply beyond the threshold.

  • Complexity: The classical cost of calculating the MPDs scales linearly with D, but intermediate virtual dimensions grow as χd^k after each disentangling step.The algorithm therefore imposes an upper cutoff ˜χ and truncates dimensions that exceed it.
  • Error propagation: For D > log_d ˜χ, the NLF suddenly increases because truncations made during encoding and circuit evolution propagate in opposite directions.Avoiding this propagation classically requires an exponentially large cutoff ˜χ ∼ d^D.
  • Error control: For D ≤ log_d ˜χ, truncation errors remain controlled and FD decays with circuit depth as expected.In this regime, the evolved state does not yet require truncation at its final virtual dimension.
  • Complexity: The exponential cutoff requirement is a classical-simulation issue that the authors state does not occur for quantum computations.The distinction concerns simulation complexity rather than the circuit’s gate construction.

V. SUMMARY

The algorithm accurately encodes highly entangled, large-virtual-dimension MPS into quantum circuits that evolve a product state to the target state efficiently.

  • The deep quantum circuit accurately evolves a product state into targeted MPS with large virtual dimensions and/or system size.The tested MPS describe nearly gapless Hamiltonians and possess large entanglement, making existing encoding methods difficult to use.

APPENDIX: PROOF OF MATRIX PRODUCT DISENTANGLER BEING UNITARY

The appendix proves the matrix product disentangler is unitary by combining orthogonal MPS tensors with orthonormal kernel-space bases. These local orthogonality conditions yield global unitarity of the circuit operator.

  • The proof uses N = 8 as an example and states that the argument generalizes to any N.The construction relies on unitary conditions of the orthogonal MPS and orthogonal properties in the kernel space.
  • The middle gates are unitary because their bases combine an isometric MPS-tensor basis with orthonormal kernel-space bases.The kernel-space bases are orthogonal to the MPS-tensor basis, so the combined basis is orthonormal.
  • The first gate is also unitary because its MPS tensor is normalized as the orthogonal center and its remaining kernel basis is orthonormal.This gives G[1]†G[1] = I.
  • U†U = I together with U being a square d^N × d^N matrix implies UU† = I, establishing that the full operator U is unitary.The local gate conditions therefore extend to the complete circuit operator.
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