Hamiltonian Simulation by Uniform Spectral Amplification

TL;DR

This work develops a unified framework for Hamiltonian simulation that exploits structure in both the Hamiltonian and its quantum-oracle encoding. By formulating uniform spectral amplification and leveraging quantum signal processing and amplitude multiplication, the authors derive improved query complexities for simulating d-sparse and general structured Hamiltonians, including tight lower bounds. They present two complementary approaches: (i) uniform amplification via polynomial transforms of the standard-form encoding, and (ii) amplification based on overlap-state representations that exploit factored oracle structures. A key insight is the universality of the standard-form encoding, establishing equivalences between simulation and measurement and allowing non-standard approaches to be recast within the standard-form framework with modest overhead. The results offer polynomial improvements in the presence of additional structure and set the stage for more practical quantum simulations by reducing normalization factors and overhead while preserving spectral distortions within controllable bounds.

Abstract

The exponential speedups promised by Hamiltonian simulation on a quantum computer depends crucially on structure in both the Hamiltonian $\hat{H}$, and the quantum circuit $\hat{U}$ that encodes its description. In the quest to better approximate time-evolution $e^{-i\hat{H}t}$ with error $ε$, we motivate a systematic approach to understanding and exploiting structure, in a setting where Hamiltonians are encoded as measurement operators of unitary circuits $\hat{U}$ for generalized measurement. This allows us to define a \emph{uniform spectral amplification} problem on this framework for expanding the spectrum of encoded Hamiltonian with exponentially small distortion. We present general solutions to uniform spectral amplification in a hierarchy where factoring $\hat{U}$ into $n=1,2,3$ unitary oracles represents increasing structural knowledge of the encoding. Combined with structural knowledge of the Hamiltonian, specializing these results allow us simulate time-evolution by $d$-sparse Hamiltonians using $\mathcal{O}\left(t(d \|\hat H\|_{\text{max}}\|\hat H\|_{1})^{1/2}\log{(t\|\hat{H}\|/ε)}\right)$ queries, where $\|\hat H\|\le \|\hat H\|_1\le d\|\hat H\|_{\text{max}}$. Up to logarithmic factors, this is a polynomial improvement upon prior art using $\mathcal{O}\left(td\|\hat H\|_{\text{max}}+\frac{\log{(1/ε)}}{\log\log{(1/ε)}}\right)$ or $\mathcal{O}(t^{3/2}(d \|\hat H\|_{\text{max}}\|\hat H\|_{1}\|\hat H\|/ε)^{1/2})$ queries. In the process, we also prove a matching lower bound of $Ω(t(d\|\hat H\|_{\text{max}}\|\hat H\|_{1})^{1/2})$ queries, present a distortion-free generalization of spectral gap amplification, and an amplitude amplification algorithm that performs multiplication on unknown state amplitudes.

Figures (4)

• Figure 1: Dependencies of new results.
• Figure 2: (top) Circuit diagram for the phased qubiterate $\hat{W}_\phi$ constructed by the qubitization of a standard-form encoding-$(\hat{H},1,(\hat{G}^\dag\otimes \hat{I}_s)\hat{U}(\hat{G}\otimes\hat{I}_s),d)$ from Low2016hamiltonian. The phased qubiterate $\hat{W}_\phi$ encodes $\hat{H}$ in standard-form-$(\hat{H},1,\hat{W}_\phi,2d)$. Note that $\widehat{\text{Had}}$ is the Hadamard gate, $\hat{\sigma}_x$ is a single-qubit NOT gate, and we define the reflection $\widehat{\text{Ref}}_{\alpha,|0\rangle_a|0\rangle_b}=\hat{I}_{ab}-(1-e^{-i\alpha})|0\rangle\langle0|_a\otimes|0\rangle\langle0|_b$. The gate complexity of $\hat{W}_\phi$ is $\mathcal{O}(\log{(d)})$. (bottom) Circuit diagram for the composite qubiterate $\hat{W}_{\vec{\phi}}$ that encodes a standard-form-$(A[\hat{H}]+iB[\hat{H}],1,\hat{W}_{\vec{\phi}},2d)$. The query complexity of $\hat{W}_{\vec{\phi}}$ is $N$ to $\hat{G}$, controlled-$\hat{U}$, and their inverses. Its gate complexity is $\mathcal{O}(N\log{(d)})$.
• Figure 3: (top) Circuit diagram for the flexible qubiterate $\hat{V}'_\phi=|0\rangle\langle0|_c\otimes\hat{W}_{\phi}+|1\rangle\langle1|_c\otimes\hat{W}_{-\phi}$, where $\hat{\text{R}}=\hat{I}_{ab}-(1-e^{-i\phi})|0\rangle\langle0|_a\otimes|0\rangle\langle0|_b$. (bottom) Circuit diagram for the flexible composite qubiterate $\hat{V}_{\vec{\phi}}$ used to encode a standard-form-$(B[\hat{H}],1,\hat{V}_{\vec{\phi}},4d)$. The query complexity of $\hat{V}_{\vec{\phi}}$ is $N$ to $\hat{G}$, controlled-$\hat{U}$, and their inverses. Its gate complexity is $\mathcal{O}(N\log{(d)})$.
• Figure 4: (top) Circuit diagram for amplitude amplification. (middle) Circuit diagram for ampliutude amplification by phase-matching. (bottom) Circuit diagram for amplitude amplification $\hat{V}_{\vec{\phi}}$ by quantum signal processing. Note that we abbreviate the reflection operators as $\hat{\text{R}}$ and drop the state subscript here. The query complexity in all cases is $N=2n+1$, and the gate complexity is $\mathcal{O}(N\log{(d)})$.

Theorems & Definitions (64)

• Claim 1: Sparse Hamiltonian simulation
• Definition 1: Standard-form matrix encoding
• Theorem 1: Hamiltonian simulation by qubitization Thm. 1 Low2016hamiltonian
• Theorem 2: Uniform spectral amplification by spectral multiplication
• Theorem 3: Uniform spectral amplification of low-energy subspaces
• Theorem 4: Flexible quantum signal processing
• Definition 2: Sparse matrix oracles Berry2012
• Theorem 5: Sparse Hamiltonian simulation by amplified state overlap
• Theorem 6
• Theorem 7: Flexible amplitude amplification