Hamiltonian Simulation by Qubitization

TL;DR

This work introduces qubitization and the quantum signal processor to perform Hamiltonian simulation with optimal query complexity. By encoding Hermitian H as $\hat{H}=\langle G|\hat{U}|G\rangle$ in standard-form, and using an invariant SU(2) iterate, it enables general operator functions $f[\hat{H}]$, notably $e^{-i\hat{H}t}$, with cost $O(t+\log(1/\epsilon))$ and only a few extra ancilla qubits. It unifies input models including linear combinations of unitaries, sparse Hamiltonians, and purified density matrices, while extending to normal operators and broader signal-processing tasks via polynomial approximations and phase-controlled iterates. The framework provides a flexible, scalable path to practical quantum simulations and other quantum algorithms that rely on non-unitary signal processing embedded in unitary operations.

Abstract

We present the problem of approximating the time-evolution operator $e^{-i\hat{H}t}$ to error $ε$, where the Hamiltonian $\hat{H}=(\langle G|\otimes\hat{\mathcal{I}})\hat{U}(|G\rangle\otimes\hat{\mathcal{I}})$ is the projection of a unitary oracle $\hat{U}$ onto the state $|G\rangle$ created by another unitary oracle. Our algorithm solves this with a query complexity $\mathcal{O}\big(t+\log({1/ε})\big)$ to both oracles that is optimal with respect to all parameters in both the asymptotic and non-asymptotic regime, and also with low overhead, using at most two additional ancilla qubits. This approach to Hamiltonian simulation subsumes important prior art considering Hamiltonians which are $d$-sparse or a linear combination of unitaries, leading to significant improvements in space and gate complexity, such as a quadratic speed-up for precision simulations. It also motivates useful new instances, such as where $\hat{H}$ is a density matrix. A key technical result is `qubitization', which uses the controlled version of these oracles to embed any $\hat{H}$ in an invariant $\text{SU}(2)$ subspace. A large class of operator functions of $\hat{H}$ can then be computed with optimal query complexity, of which $e^{-i\hat{H}t}$ is a special case.

Figures (1)

• Figure 1: (Left) Approximation error $\epsilon=\max_{\theta\in\mathbb{R}}|A[\theta]-iC[\theta]-e^{i t \sin{(\theta)}}|$. $(A[\theta],C[\theta])$ are real Fourier series in $(\cos{(k\theta)},\sin{(k\theta)}),\;k=0,...,Q/2$, and $\epsilon$ is plotted for the upper bound $\frac{4t^q}{2^q q!}$ (blue), truncation $\sum^{\infty}_{k=q} 2|J_{k}[t]|$ (black), and best possible karam1999 (red), for $Q=2,4,8,16,32,64,\infty$ queries to the controlled iterate $\hat{W}$, where $q=1+Q/2$. (Right) Queries per unit of simulation time to unitary $\hat{U}$ encoding $\hat{H}=\langle G|\hat{U}|G\rangle$ at target error $\|\langle+|\hat{V}_{\vec{\varphi}}|+\rangle-e^{-i\hat{H}t}\|=10^{-2,-4,-8,-16}$ for the BCCKS algorithm (thin), and this work using the truncated approximation to $e^{i t \sin{(\theta)}}$ (thick).

Theorems & Definitions (33)

• Theorem 1: Optimal Hamiltonian simulation by Qubitization
• Definition 1: Standard-form
• Definition 2: Quantum Signal Processor
• Theorem 2: Qubitization
• Theorem 3: Ancilla-Free Quantum Signal Processing
• Theorem 4: Single-Ancilla Quantum Signal Processing
• Lemma 5: Standard-Form Encoding by a Linear Combination of Unitaries
• proof
• Lemma 6: Standard-Form Encoding of a $d$-Sparse Hamiltonian
• proof