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Faster State Preparation with Randomization

Yue Wang, Xiao-Ming Zhang, Xiao Yuan, Qi Zhao

TL;DR

A randomized protocol is introduced that fundamentally improves the accuracy-cost trade-off for states with hierarchical amplitude structures, where amplitudes decay exponentially or by a power law with exponent greater than one.

Abstract

Quantum state preparation remains a dominant cost in many quantum algorithms. We introduce a randomized protocol that fundamentally improves the accuracy-cost trade-off for states with hierarchical amplitude structures, where amplitudes decay exponentially or by a power law with exponent greater than one. Rather than commonly employed deterministically truncating small amplitudes, we prepare an ensemble of simple circuits: each retains all large amplitudes while amplifying a single small one. We rigorously prove that the randomized ensemble achieves a quadratic improvement in trace-distance error over the deterministic truncation method. This quadratic scaling halves the number of encoded amplitudes for exponential decay and yields polynomial reductions for power-law decay, which directly translates to a reduction in circuit depth. Demonstrations on molecular wavefunctions (LiH), many-body ground states (transverse-field Ising), and machine-learning parameters (ResNet) validate the approach across diverse applications. The method broadens the class of states preparable on near-term and fault-tolerant quantum devices.

Faster State Preparation with Randomization

TL;DR

A randomized protocol is introduced that fundamentally improves the accuracy-cost trade-off for states with hierarchical amplitude structures, where amplitudes decay exponentially or by a power law with exponent greater than one.

Abstract

Quantum state preparation remains a dominant cost in many quantum algorithms. We introduce a randomized protocol that fundamentally improves the accuracy-cost trade-off for states with hierarchical amplitude structures, where amplitudes decay exponentially or by a power law with exponent greater than one. Rather than commonly employed deterministically truncating small amplitudes, we prepare an ensemble of simple circuits: each retains all large amplitudes while amplifying a single small one. We rigorously prove that the randomized ensemble achieves a quadratic improvement in trace-distance error over the deterministic truncation method. This quadratic scaling halves the number of encoded amplitudes for exponential decay and yields polynomial reductions for power-law decay, which directly translates to a reduction in circuit depth. Demonstrations on molecular wavefunctions (LiH), many-body ground states (transverse-field Ising), and machine-learning parameters (ResNet) validate the approach across diverse applications. The method broadens the class of states preparable on near-term and fault-tolerant quantum devices.
Paper Structure (3 sections, 4 theorems, 47 equations, 2 figures, 1 algorithm)

This paper contains 3 sections, 4 theorems, 47 equations, 2 figures, 1 algorithm.

Table of Contents

  1. State-Level Mixing Lemma
  2. Trace Norm Error Bound
  3. Configuration Interaction and Implementation

Key Result

Lemma 1

Let $\ket{\psi}$ be a target pure quantum state, and let $\{\ket{\psi_m}\}_m$ be a family of pure states with associated probabilities $\{p_m\}_m$ satisfying: $\|\ket{\psi_m} - \ket{\psi}\| \le a, \forall m,$ and $\left\| \sum_m p_m \ket{\psi_m} - \ket{\psi} \right\| \le b,$ for some $a, b > 0$. Def

Figures (2)

  • Figure 1: CI coefficients for LiH: trace-norm distance versus threshold comparing randomized and cutoff reconstructions. As the threshold decreases and fewer terms are discarded, both errors drop, with the randomized method exhibiting a quadratically faster reduction. The inset shows the reduction in the number of coefficients. At a target error of $10^{-4}$, the number of kept coefficients is reduced by 53%. Further numerical results for (i) a state built from a simplified ResNet's parameters using 10 qubits and (ii) the TFIM ground state at $N=11$ with $J=h=1.0$ illustrate the accuracy advantages across diverse settings.
  • Figure 2: Coefficient distribution of the CI state for LiH molecule.

Theorems & Definitions (7)

  • Lemma 1
  • Theorem 1
  • Lemma 2
  • proof
  • Definition 1: Large-/Small-Amplitude Partition
  • Theorem 2: Random mixing error bound
  • proof