Quantum Advantage in Resource Estimation

TL;DR

The paper identifies the estimation of quantum algorithm resources as a prime candidate for quantum advantage by directly measuring simulation errors on quantum hardware. It introduces a Hadamard-test-based method to estimate the Trotter error $\epsilon_k^\text{TS}$ for $p$-order Suzuki-Trotter decompositions via a state-phase metric $\theta_\psi$, enabling more accurate resource estimates than classical bounds. Results show that using a higher-order reference (e.g., $p'=4$) yields <1% relative error in phase estimates and can reduce QPE resource costs by several orders of magnitude, suggesting feasibility for MegaQuop-scale devices and even NISQ-era demonstrations. The work highlights a practical pathway for demonstrating quantum advantage in building and simulating large quantum computers, with potential extensions to other error sources and broader quantum simulation tasks.

Abstract

Quantum computing promises the ability to compute properties of quantum systems exponentially faster than classical computers. Quantum advantage is achieved when a practical problem is solved more efficiently on a quantum computer than on a classical computer. Demonstrating quantum advantage requires a powerful quantum computer with low error rates and an efficient quantum algorithm that has a useful application. Despite rapid progress in hardware development, we still lack useful applications that are feasible for the next generation of quantum computers. Here we argue that an exponential quantum advantage exists in producing numerical resource estimates of larger quantum algorithms by accurately measuring simulation errors. We provide a quantum algorithm for measuring simulation errors of Trotter-based algorithms. Our results indicate that this method will reduce runtimes of quantum algorithms by approximately three orders of magnitude for one-hundred qubit systems. We also predict that these reductions will increase with system size. The methods we propose require relatively few qubits and operations, meaning the next generation of quantum computers could compute simulation errors for classically intractable systems. Since the underlying computations that lead to reduced resource estimates are infeasible for classical computers, this task is a candidate for demonstrating practical quantum advantage.

Figures (11)

• Figure 1: Resource Estimation Workflow Resource estimation begins by formulating the problem and defining the objective of the computation. An algorithm is chosen that performs the desired computation and an analysis of the error sources in the simulation is performed. The magnitude of the simulation error is estimated using either classical or quantum computers. The estimated simulation error results in an implementation of the algorithm with fixed quantum resources; these estimates determine the size of the quantum computer required to solve the problem.
• Figure 2: Hadamard Test for Computing Phase Error A Hadamard test is a quantum algorithm for amplitude estimation. a, The Hadamard test computing the real component of the amplitude, $\bra{\psi} U_\text{ref}^\dagger(t) U_{H^\prime}(t) \ket{\psi}$, is shown. b, The circuit for evaluating the imaginary component of the amplitude is shown.
• Figure 3: Approximating Simulation Errora, The relative error on the approximate phase error ($\tilde{\theta}_\psi$) is shown as a function of the Suzuki-Trotter order of the reference approximation ($p^\prime$). Results are shown for the ground state ($\epsilon_0^\text{TS}$) of chemistry Hamiltonians (blue circles) and random Pauli Hamiltonians (green squares). The target approximation is a second-order Suzuki-Trotter decomposition ($p=2$) with $t = \pi / ||4 H||$. The red line depicts a relative error of $1\%$. Two chemistry systems with $|\theta_\psi| < 10^{-10}$ are omitted since floating point errors obscure our numerics. b, The relative error on the approximate phase error ($\tilde{\theta}_\psi$) is shown as a function of system size ($N$). Results are shown for the ground state ($\epsilon_0^\text{TS}$) of chemistry Hamiltonians (blue circles) and random Pauli Hamiltonians (green squares). The target approximation is a second-order Suzuki-Trotter decomposition ($p=2$) and the reference approximation is a fourth-order Suzuki-Trotter decomposition ($p^\prime = 4$) with $t = \pi / ||4 H||$. The red line depicts a relative error of $1\%$. c, The accuracy of the approximations to the Trotter error on the ground state ($\epsilon_0^\text{TS}$) is shown as a function of the system size ($N$). The error ratio is calculated as $|\epsilon^* / \epsilon_0^\text{TS}|$ where $\epsilon^*$ represents either the classical commutator bounds (Eq. \ref{['eq:classical-commutator-bounds']}) or the quantum phase error approximation (Eq. \ref{['eq:phase-approximation-error']}). The orange upward (red downward) triangles correspond to the classical error estimates and the blue circles (green squares) correspond to the quantum error estimates for chemistry systems (random Pauli Hamiltonians). The target approximation is a second-order Suzuki-Trotter decomposition ($p=2$) and the reference approximation is a fourth-order Suzuki-Trotter decomposition ($p^\prime = 4$) with $t = \pi / ||4 H||$. The dashed black line denotes an error ratio of $1$.
• Figure 4: Resource Estimatesa, The number of operations ($O$) required to compute the ground state energy using QPE is shown as a function of system size. The total error budget of QPE is set to $\epsilon = 0.0016$ Hartree ($\epsilon = 0.01$) for the for the chemistry systems (random Pauli Hamiltonians). The orange upward (red downward) triangles correspond to the resource estimates produced using classical computers and the blue circles (green squares) correspond to the resource estimates produced using quantum computers for chemistry systems (random Pauli Hamiltonians). The gray crosses correspond to resource estimates produced using the Trotter error on the ground state ($\epsilon_0^\text{TS}$). The shaded yellow (red) regions depict the systems for which we could compute ground state energies on MegaQuop (NISQ) machines. We set the upper limit of MegaQuop (NISQ) machines at fewer than $10^8$ ($10^5$) operations. b, The number of operations required to compute the simulation error ($\tilde{\theta}_{\psi}$) is shown as a function of system size. These estimates assume the target approximation is a $p = 2$ Suzuki-Trotter decomposition and the reference approximation is a $p^\prime = 4$ Suzuki-Trotter decomposition. The blue circles (green squares) correspond to resource estimates for chemistry systems (random Pauli Hamiltonians). The shaded yellow (red) regions depict the systems for which we could compute the simulation error using MegaQuop (NISQ) machines. Operations are counted as the number of time evolutions of multi-qubit Pauli operators.
• Figure 5: Trotter-Based Quantum Phase Estimationa, A quantum circuit for QPE using an approximate time evolution unitary is shown. We refer to the qubits beginning in the $\ket{0}$ state as the phase register and the number of these qubits is given by $B$. b, Applying effective time evolution operations of durations $St$ is achieved by $S$ applications of the effective time evolution unitary for time $t$.