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Fast quantum computation with all-to-all Hamiltonians

Chao Yin

TL;DR

This work analyzes the computational power of time-dependent all-to-all Hamiltonians and demonstrates two fundamental speedups over conventional circuit-based computation. First, any two-qubit gate can be simulated in time $T=\mathrm{O}(1/N)$ with polynomially small error, enabling rapid preparation of entangled states such as GHZ and W and toffoli-type operations; second, any depth-$D$ quantum circuit can be simulated in time $T=\mathrm{O}(D/\sqrt{N})$ with constant space overhead and polynomially small error using a randomized protocol. The key technique treats ancilla qubits as a bosonic mode (via the Dicke manifold and Holstein-Primakoff mapping), employing spin squeezing, displacement, and a Mølmer-Sørensen–like interaction to mediate non-commuting, all-to-all couplings, and then extends to a layer-wise parallelism through Fourier focusing to achieve the speedups. These results align with Lieb-Robinson bounds for power-law interactions, showing the speedups are near-optimal in the given setting and highlighting a distinct regime where Hamiltonian dynamics outperform circuit-based planning. The work opens directions for experimental realizations, error-correction under Hamiltonian evolution, and possible faster or deterministic speedups, strengthening the bridge between many-body physics and quantum computation.

Abstract

All-to-all interactions arise naturally in many areas of theoretical physics and across diverse experimental quantum platforms, motivating a systematic study of their information-processing power. Assuming each pair of qubits interacts with $\mathrm{O}(1)$ strength, time-dependent all-to-all Hamiltonians can simulate arbitrary all-to-all quantum circuits, performing quantum computation in time proportional to the circuit depth. We show that this naive correspondence is far from optimal: all-to-all Hamiltonians can process information on much shorter timescales. First, we prove that any two-qubit gate can be simulated by all-to-all Hamiltonians on $N$ qubits in time $\mathrm{O}(1/N)$ (up to factor $N^δ$ with an arbitrarily small constant $δ>0$), with polynomially small error $1/\mathrm{poly}(N)$. Immediate consequences include: 1) Certain $\mathrm{O}(N)$-qubit unitaries and entangled states, such as the multiply-controlled Toffoli gate and the GHZ and W states, can be generated in $\mathrm{O}(1/N)$ time; 2) Trading space for time, any quantum circuit can be simulated in arbitrarily short time; 3) Information could propagate in a fast way that saturates known Lieb-Robinson bounds in strongly power-law interacting systems. Our second main result proves that any depth-$D$ quantum circuit can be simulated by a randomized Hamiltonian protocol in time $T=\mathrm{O}(D/\sqrt{N})$, with constant space overhead and polynomially small error. The techniques underlying our results depart fundamentally from the existing literature on parallelizing commuting gates: We rely crucially on non-commuting Hamiltonians and draw on diverse physical ideas.

Fast quantum computation with all-to-all Hamiltonians

TL;DR

This work analyzes the computational power of time-dependent all-to-all Hamiltonians and demonstrates two fundamental speedups over conventional circuit-based computation. First, any two-qubit gate can be simulated in time with polynomially small error, enabling rapid preparation of entangled states such as GHZ and W and toffoli-type operations; second, any depth- quantum circuit can be simulated in time with constant space overhead and polynomially small error using a randomized protocol. The key technique treats ancilla qubits as a bosonic mode (via the Dicke manifold and Holstein-Primakoff mapping), employing spin squeezing, displacement, and a Mølmer-Sørensen–like interaction to mediate non-commuting, all-to-all couplings, and then extends to a layer-wise parallelism through Fourier focusing to achieve the speedups. These results align with Lieb-Robinson bounds for power-law interactions, showing the speedups are near-optimal in the given setting and highlighting a distinct regime where Hamiltonian dynamics outperform circuit-based planning. The work opens directions for experimental realizations, error-correction under Hamiltonian evolution, and possible faster or deterministic speedups, strengthening the bridge between many-body physics and quantum computation.

Abstract

All-to-all interactions arise naturally in many areas of theoretical physics and across diverse experimental quantum platforms, motivating a systematic study of their information-processing power. Assuming each pair of qubits interacts with strength, time-dependent all-to-all Hamiltonians can simulate arbitrary all-to-all quantum circuits, performing quantum computation in time proportional to the circuit depth. We show that this naive correspondence is far from optimal: all-to-all Hamiltonians can process information on much shorter timescales. First, we prove that any two-qubit gate can be simulated by all-to-all Hamiltonians on qubits in time (up to factor with an arbitrarily small constant ), with polynomially small error . Immediate consequences include: 1) Certain -qubit unitaries and entangled states, such as the multiply-controlled Toffoli gate and the GHZ and W states, can be generated in time; 2) Trading space for time, any quantum circuit can be simulated in arbitrarily short time; 3) Information could propagate in a fast way that saturates known Lieb-Robinson bounds in strongly power-law interacting systems. Our second main result proves that any depth- quantum circuit can be simulated by a randomized Hamiltonian protocol in time , with constant space overhead and polynomially small error. The techniques underlying our results depart fundamentally from the existing literature on parallelizing commuting gates: We rely crucially on non-commuting Hamiltonians and draw on diverse physical ideas.

Paper Structure

This paper contains 8 sections, 5 theorems, 8 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $1/N$-time protocols
  4. Implications
  5. Proof sketch
  6. $\sqrt{N}$ speedup for any quantum circuit
  7. Proof sketch
  8. Outlook

Key Result

Theorem 1

For any constants $\delta_{\rm T}\in(0,1)$ and locality $K\ge 2$, the following holds for sufficiently large $N$. For a set of data qubits $\{0\}\cup S$, there exists a Hamiltonian protocol $H(t)$ that simulates a bunch of CZ gates $\prod_i\mathrm{CZ}_{0,i\in S}$ with polynomially small error for any $\ket{\psi}$ on data qubits, in total evolution time Here $\mathcal{T}$ is time-ordering.

Figures (2)

  • Figure 1: Sketch of the idea for Theorem \ref{['thm:CZ']}. (a) We mediate data-qubit interactions by $N$ ancila qubits, which simulates a boson mode with boson number $b^\dagger b\lesssim N$. (b) The protocol first amplifies the signal $Z_0$ to boson position $\hat{x}$ (red $\rightarrow$ yellow $\rightarrow$ green $\rightarrow$ blue, where solid/dashed contours correspond to $Z_0=\pm 1$), then engineers a potential $\pm V(\hat{x})$ controlled by the target data qubits $S$, and finally reverses the signal amplification step.
  • Figure 2: Sketch of the idea for Theorem \ref{['thm:circ']}. (a) An exact $T=\mathrm{O}(1/\sqrt{N})$ protocol of simulating one gate $\mathrm{CZ}_{0,1}$. The four colored trajectories correspond to the four choices of $Z_0,Z_1=\pm1$. (b) We Fourier transform the ancilla qubits $X_i^+\rightarrow \widetilde{X}_k^+$, which focus the coupling strengths to make them individually stronger. Although the Fourier modes $\widetilde{X}_k^+$ are not simple spins or bosons, they can be approximated so for almost all inputs of our simulation protocol.

Theorems & Definitions (5)

  • Theorem 1
  • Corollary 2
  • Corollary 3
  • Corollary 4
  • Theorem 5