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Randomization Accelerates Series-Truncated Quantum Algorithms

Yue Wang, Qi Zhao

TL;DR

The paper tackles the high resource cost of quantum algorithms built on truncated series by introducing Randomized Truncated Series (RTS), a framework that mixes two truncated polynomial approximations with a tunable probability to cancel truncation errors. RTS achieves quadratic suppression of truncation errors and enables a continuous, fractional effective truncation order, reducing the average circuit size across a broad class of algorithms. The authors prove a general error bound for the mixed channel and demonstrate RTS in Hamiltonian simulation (via LCU in BCCKS), Quantum Signal Processing (including HS and USA), and ODE solvers, accompanied by numerical results showing substantial gate savings. This framework provides a versatile, practical path toward more feasible quantum advantage in near- to mid-term applications by lowering resource demands without sacrificing accuracy. The results suggest RTS can be extended to non-unitary dynamics and time-dependent problems, further broadening its impact on quantum algorithm design.

Abstract

Quantum algorithms typically demand prohibitively complicated circuits to solve practical problems. Previous studies have shown that classical randomness can accelerate some specific quantum algorithms. In this work, we introduce the Randomized Truncated Series (RTS) which extends this acceleration to all quantum algorithms that rely on truncated series approximations. RTS offers two key advantages: it quadratically suppresses truncation errors and allows for continuous adjustment of the effective truncation order. By leveraging random mixing between two quantum circuits, RTS ensures that their probabilistic combination accurately realizes the desired algorithm, while significantly reducing the average circuit size. We demonstrate the versatility of RTS through concrete applications. Our results shed light on the path toward practical quantum advantage.

Randomization Accelerates Series-Truncated Quantum Algorithms

TL;DR

The paper tackles the high resource cost of quantum algorithms built on truncated series by introducing Randomized Truncated Series (RTS), a framework that mixes two truncated polynomial approximations with a tunable probability to cancel truncation errors. RTS achieves quadratic suppression of truncation errors and enables a continuous, fractional effective truncation order, reducing the average circuit size across a broad class of algorithms. The authors prove a general error bound for the mixed channel and demonstrate RTS in Hamiltonian simulation (via LCU in BCCKS), Quantum Signal Processing (including HS and USA), and ODE solvers, accompanied by numerical results showing substantial gate savings. This framework provides a versatile, practical path toward more feasible quantum advantage in near- to mid-term applications by lowering resource demands without sacrificing accuracy. The results suggest RTS can be extended to non-unitary dynamics and time-dependent problems, further broadening its impact on quantum algorithm design.

Abstract

Quantum algorithms typically demand prohibitively complicated circuits to solve practical problems. Previous studies have shown that classical randomness can accelerate some specific quantum algorithms. In this work, we introduce the Randomized Truncated Series (RTS) which extends this acceleration to all quantum algorithms that rely on truncated series approximations. RTS offers two key advantages: it quadratically suppresses truncation errors and allows for continuous adjustment of the effective truncation order. By leveraging random mixing between two quantum circuits, RTS ensures that their probabilistic combination accurately realizes the desired algorithm, while significantly reducing the average circuit size. We demonstrate the versatility of RTS through concrete applications. Our results shed light on the path toward practical quantum advantage.
Paper Structure (19 sections, 13 theorems, 105 equations, 2 figures, 1 table)

This paper contains 19 sections, 13 theorems, 105 equations, 2 figures, 1 table.

Table of Contents

  1. Results
  2. Applications
  3. BCCKS example
  4. Quantum Signal Processing (QSP) examples
  5. Ordinary Differential Equations example
  6. Numerical result
  7. Discussion
  8. Methods
  9. BCCKS
  10. QSP
  11. Hamiltonian Simulation
  12. Uniform spectral amplification (USA)
  13. ODE
  14. Framework implementation on BCCKS algorithm
  15. Proof of Corollary \ref{['cor:LCU']}
  16. ...and 4 more sections

Key Result

Lemma 1

Let $V_1$ and $V_2$ be near-unitary operators approximating an ideal operator $U$. Denote the operator $V_{m} := pV_1 + (1-p)V_2$. Assume the operator norm follows $\|V_1-U \|\le a_1$, $\|V_2-U \|\le a_2$, and $\|V_{m}-U\|\le b$, then the density operator $\rho = \ket{\psi}\bra{\psi}$ acted on by th where $\varepsilon= 4b+2pa_1^2+2(1-p)a_2^2$, $\mathcal{U}(\rho) = U\rho U^{\dagger}$ and $\|\cdot\|

Figures (2)

  • Figure 1: Illustration of conceptual idea. We denote $f_k$ as the $k$-th term in $F(H)$. (a) depicts the conventional approach. The target error $\epsilon$ falls between truncation errors for two series of order $K$ and $K+1$, resulting in inefficiency in the $(K+1)$-th term. (b) demonstrates the RTS method, where we mix two series expansions $F_1$ (depicted in blue brackets) and $F_2$ (depicted in orange brackets) with probability $p$ and $1-p$, respectively. $\tilde{f}_k = 1/(1-p) f_k$ in the orange bracket are modified terms that return to $f_k$ after sampling the measurement result. The amplification coefficient $1/(1-p)$ will be reverted to unity because its contribution to the final result is suppressed by its mixing probability $1-p$. Consequently, the output of RTS includes information on higher-order terms in the series expansion, and better approximates $F(H)$.
  • Figure 2: (a) Performance enhancement achieved by applying RTS to the BCCKS algorithm. We denote $V_{1,K1}$ as the performance of the BCCKS algorithm with truncation order $K_1$. With RTS, the overall error exhibits a substantial reduction of several orders of magnitude, consistent with quadratic error suppression. Each point on the curve represents the error obtained using the optimal set of parameters $\{K_1,K_2,p\}$. $\tilde{G} = 131574240$ is a multiplier for CNOT gate cost. (b) illustrates the variation of epsilon with $K_1$ and $K_2$ for a fixed $p = 0.8$.

Theorems & Definitions (26)

  • Definition 1
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • Corollary 4
  • Lemma 2
  • ...and 16 more