arXiv subjects

Guang Hao Low

Publications and source records attributed to Guang Hao Low.

At least 19 recordsLinked to original sources

arXiv2026

Low-gate-count block encodings for second-quantized fermionic Hamiltonians

Efficient block encoding of many-body Hamiltonians is a central requirement for quantum algorithms in scientific computing, particularly in the early fault-tolerant era. In this work, we introduce new explicit constructions for block encoding second-quantized Hamiltonians that substantially reduce Clifford+T gate complexity and ancilla overhead. By utilizing a data lookup strategy based on the SWAP architecture for the sparsity oracle $O_C$, and a direct sampling method for the amplitude oracle $O_A$ with SELECT-SWAP architecture, we achieve a T count that scales as $\mathcal{\tilde{O}}(\sqrt{L})$ with respect to the number of interaction terms $L$ in general second-quantized Hamiltonians. We also achieve an improved constant factor in the Clifford gate count of our oracle. Furthermore, we design a block encoding that directly targets the $η$-particle subspace, thereby reducing the subnormalization factor from $\mathcal{O}(L)$ to $\mathcal{O}(\sqrt{L})$, and improving fault-tolerant efficiency when simulating systems with fixed particle numbers. Building on the block encoding framework developed for general many-body Hamiltonians, we extend our approach to electronic Hamiltonians whose coefficient tensors exhibit translation invariance or possess decaying structures. Our results provide a practical path toward early fault-tolerant quantum simulation of many-body systems, substantially lowering resource overheads compared to previous methods.

quant-ph↗

arXiv2026

Near-frustration-free electronic structure Hamiltonian representations and lower bound certificates

Hamiltonian representations based on the sum-of-squares (SOS) hierarchy provide rigorous lower bounds on ground-state energies and facilitate the design of efficient classical and quantum simulation algorithms. This work presents a unified framework connecting SOS decompositions with variational two-particle reduced density matrix (v2RDM) theory. We demonstrate that the ``weighted'' SOS ansatz naturally recovers the dual of the v2RDM program, enabling the strict enforcement of symmetry constraints such as particle number and spin. We provide explicit SOS constructions for the Hubbard model and electronic structure Hamiltonians, ranging from spin-free approximations to full rank-2 expansions. We also highlight theoretical connections to block-invariant symmetry shifts. Numerical benchmarks on molecular systems and Iron-Sulfur clusters validate these near frustration-free representations, demonstrating their utility in improving spectral gap amplification and reducing block encoding costs in quantum algorithms.

quant-ph↗

arXiv2026

Spectral amplification for ground-state energy estimation of electronic structure in first quantization

We demonstrate an asymptotic gate complexity improvement in first-quantized ground-state energy estimation of electronic structure Hamiltonians in a plane wave basis by employing the sum-of-squares spectral gap amplification protocol. The improvement relies on identifying a sum-of-squares representation of the Hamiltonian which provides a lower bound certificate and low cost block encoding that leads to a provably lower quantum phase estimation gate cost. This is achieved by using a sum-of-squares operator generated by the total charge density operator resulting in a block encoding normalization improvement of $λ= \mathcal{O}\left(ηΔ^{-1.5}+η^{1.5}Δ^{-1} \right)$ compared to prior work $λ= \mathcal{O}(ηΔ^{-2}+η^2Δ^{-1})$ where $η$ is the number of electrons and $Δ$ is the simulation grid spacing. The asymptotic reduction in block encoding normalization and similar block encoding costs to prior work is demonstrated to reduce resource estimates for materials and chemical systems by a factor of $2 - 44\times$ corresponding to the lowest cost estimates for ab initio materials simulation.

quant-ph↗

arXiv2026

A Denser Planar Surface Code

We present a quantum code implementable on a regular $2$D hex grid with an estimated encoding rate up to $4.5\times$ of that of a rotated surface code patch using circuit-level noise in a one- and two-qubit $10^{-3}$ error uniform depolarizing model. Our approach is based on yoking a dense packing of surface code twist defects, enabled by new stabilizer measurement cycles with an optimal four layers of nearest-neighbor two-qubit gates, almost no distance-reducing hook errors, and efficient decoding. We demonstrate a space-efficient architecture for computing on densely packed logical qubits, including new padding-free lattice surgery protocols in an optimal bounding box of $2d^2$ data and measurement qubits per patch. Assuming a $1μ$s surface code cycle time and a $10μ$s reaction time, these developments enable chemically accurate ground state phase estimation of a broad class of `utility-scale' electronic structure simulation problems such as the $108$ spin-orbital FeMoco-based nitrogen fixation catalyst in under a month with $89$k noisy superconducting qubits. We elucidate a Pareto frontier of space-time trade-offs and find a minimum physical quantum volume of $1.3$ mega-qubit-hours. These correspond to a $36\times$ space and $6.6\times$ spacetime improvement, respectively, over our previous state-of-the-art minimum-Toffoli resource estimates (Phys. Rev. X 15, 041016).

quant-ph↗

arXiv2026

Quantum eigenvalue processing

Many problems in linear algebra -- such as those arising from non-Hermitian physics and differential equations -- can be solved on a quantum computer by processing eigenvalues of the non-normal input matrices. However, the existing Quantum Singular Value Transformation (QSVT) framework is ill-suited for this task, as eigenvalues and singular values are different in general. We present a Quantum EigenValue Transformation (QEVT) framework for applying arbitrary polynomial transformations on eigenvalues of block-encoded non-normal operators, and a related Quantum EigenValue Estimation (QEVE) algorithm for operators with real spectra. QEVT has query complexity to the block encoding nearly recovering that of the QSVT for a Hermitian input, and QEVE achieves the Heisenberg-limited scaling for diagonalizable input matrices. As applications, we develop a linear differential equation solver with strictly linear time query complexity for average-case diagonalizable operators, as well as a ground state preparation algorithm that upgrades previous nearly optimal results for Hermitian Hamiltonians to diagonalizable matrices with real spectra. Underpinning our algorithms is an efficient method to prepare a quantum superposition of Faber polynomials, which generalize the nearly-best uniform approximation properties of Chebyshev polynomials to the complex plane. Of independent interest, we also develop techniques to generate $n$ Fourier coefficients with $\mathbf{O}(\mathrm{polylog}(n))$ gates compared to prior approaches with linear cost.

quant-ph↗

arXiv2026

End-to-End Simulation of Chemical Dynamics on a Quantum Computer

Simulations of chemical dynamics are a powerful means for understanding chemistry. However, classical computers struggle to simulate many chemical processes, especially non-adiabatic ones, where the Born-Oppenheimer approximation breaks down. Quantum computers could simulate quantum-chemical dynamics more efficiently than classical computers, but there is currently no complete quantum algorithm for calculating dynamical observables to within a known error. Here, we develop an efficient, end-to-end quantum algorithm for simulating chemical dynamics that avoids all uncontrolled approximations (including the Born-Oppenheimer approximation) and whose error is bounded subject to mild assumptions. To do so, we treat the nuclei and the electrons on an equal footing and simulate the full molecular wavefunction on a momentum-space grid in first quantization, including all algorithmic steps: initial-state preparation, time evolution using qubitization, and measurement of chemical observables such as reaction yields and rates. Our work gives the first algorithm for quantum simulation of chemistry whose end-to-end complexity achieves sublinear scaling in the size of the grid. We achieve this by developing an exponentially faster method for initial-state-preparation. Photochemistry is a likely early application of our algorithm and we estimate resources required for end-to-end simulations of non-adiabatic dynamics of atmospherically important molecules. Classically intractable photochemical computations could be performed using resources comparable to those required for other chemical applications of quantum computing.

quant-ph↗

arXiv2026

Quantum linear system algorithm with optimal queries to initial state preparation

Quantum algorithms for linear systems produce the solution state $A^{-1}|b\rangle$ by querying two oracles: $O_A$ that block encodes the coefficient matrix and $O_b$ that prepares the initial state. We present a quantum linear system algorithm making $\mathbfΘ\left(1/\sqrt{p}\right)$ queries to $O_b$, which is optimal in the success probability, and $\mathbf{O}\left(κ\log\left(1/p\right)\left(\log\log\left(1/p\right)+\log\left({1}/ε\right)\right)\right)$ queries to $O_A$, nearly optimal in all parameters including the condition number and accuracy. Notably, our complexity scaling of initial state preparation holds even when $p$ is not known $\textit{a priori}$. This contrasts with recent results achieving $\mathbf{O}\left(κ\log\left({1}/ε\right)\right)$ complexity to both oracles, which, while optimal in $O_A$, is highly suboptimal in $O_b$ as $κ$ can be arbitrarily larger than $1/\sqrt{p}$. In various applications such as solving differential equations, preparing ground states of operators with real spectra, and estimating and transforming eigenvalues of non-normal matrices, we can further improve the dependence on $p$ using a block preconditioning scheme to nearly match or outperform best previous results based on other methods, which also furnishes an extremely simple quantum linear system algorithm with an optimal query complexity to $O_A$. Underlying our results is a new Variable Time Amplitude Amplification algorithm with Tunable thresholds (Tunable VTAA), which fully characterizes generic nested amplitude amplifications, improves the $\ell_1$-norm input cost scaling of Ambainis to an $\ell_{\frac{2}{3}}$-quasinorm scaling, and admits a deterministic amplification schedule for the quantum linear system problem.

quant-ph↗

arXiv2025

Unified Architecture for Quantum Lookup Tables

Quantum access to arbitrary classical data encoded in unitary black-box oracles underlies interesting data-intensive quantum algorithms, such as machine learning or electronic structure simulation. The feasibility of these applications depends crucially on gate-efficient implementations of these oracles, which are commonly some reversible versions of the boolean circuit for a classical lookup table. We present a general parameterized architecture for quantum circuits implementing a lookup table that encompasses all prior work in realizing a continuum of optimal tradeoffs between qubits, non-Clifford gates, and error resilience, up to logarithmic factors. Our architecture assumes only local 2D connectivity, yet recovers results that previously required all-to-all connectivity, particularly, with the appropriate parameters, poly-logarithmic error scaling. We also identify novel regimes, such as simultaneous sublinear scaling in all parameters. These results enable tailoring implementations of the commonly used lookup table primitive to any given quantum device with constrained resources.

quant-ph↗

arXiv2025

The Grand Challenge of Quantum Applications

This perspective outlines promising pathways and critical obstacles on the road to developing useful quantum computing applications, drawing on insights from the Google Quantum AI team. We propose a five-stage framework for this process, spanning from theoretical explorations of quantum advantage to the practicalities of compilation and resource estimation. For each stage, we discuss key trends, milestones, and inherent scientific and sociological impediments. We argue that two central stages -- identifying concrete problem instances expected to exhibit quantum advantage, and connecting such problems to real-world use cases -- represent essential and currently under-resourced challenges. Throughout, we touch upon related topics, including the promise of generative artificial intelligence for aspects of this research, criteria for compelling demonstrations of quantum advantage, and the future of compilation as we enter the era of early fault-tolerant quantum computing.

quant-ph↗

arXiv2025

The FLuid Allocation of Surface code Qubits (FLASQ) cost model for early fault-tolerant quantum algorithms

Holistic resource estimates are essential for guiding the development of fault-tolerant quantum algorithms and the computers they will run on. This is particularly true when we focus on highly-constrained early fault-tolerant devices. Many attempts to optimize algorithms for early fault-tolerance focus on simple metrics, such as the circuit depth or T-count. These metrics fail to capture critical overheads, such as the spacetime cost of Clifford operations and routing, or miss they key optimizations. We propose the FLuid Allocation of Surface code Qubits (FLASQ) cost model, tailored for architectures that use a two-dimensional lattice of qubits to implement the two-dimensional surface code. FLASQ abstracts away the complexity of routing by assuming that ancilla space and time can be fluidly rearranged, allowing for the tractable estimation of spacetime volume while still capturing important details neglected by simpler approaches. At the same time, it enforces constraints imposed by the circuit's measurement depth and the processor's reaction time. We apply FLASQ to analyze the cost of a standard two-dimensional lattice model simulation, finding that modern advances (such as magic state cultivation and the combination of quantum error correction and mitigation) reduce both the time and space required for this task by an order of magnitude compared with previous estimates. We also analyze the Hamming weight phasing approach to synthesizing parallel rotations, revealing that despite its low T-count, the overhead from imposing a 2D layout and from its use of additional ancilla qubits will make it challenging to benefit from in early fault-tolerance. We hope that the FLASQ cost model will help to better align early fault-tolerant algorithmic design with actual hardware realization costs without demanding excessive knowledge of quantum error correction from quantum algorithmists.

quant-ph↗

arXiv2025

Optimal quantum simulation of linear non-unitary dynamics

We present a quantum algorithm for simulating the time evolution generated by any bounded, time-dependent operator $-A$ with non-positive logarithmic norm, thereby serving as a natural generalization of the Hamiltonian simulation problem. Our method generalizes the recent Linear-Combination-of-Hamiltonian-Simulation (LCHS) framework. In instances where $A$ is time-independent, we provide a block-encoding of the evolution operator $e^{-At}$ with $\mathcal{O}\big(t\log\frac{1}ε)$ queries to the block-encoding oracle for $A$. We also show how the normalized evolved state can be prepared with $\mathcal{O}(1/\|e^{-At}|{\vec{u}_0}\rangle\|)$ queries to the oracle that prepares the normalized initial state $|{\vec{u}_0}\rangle$. These complexities are optimal in all parameters and improve the error scaling over prior results. Furthermore, we show that any improvement of our approach exceeding a constant factor of approximately 3 is infeasible. For general time-dependent operators $A$, we also prove that a uniform trapezoidal rule on our LCHS construction yields exponential convergence, leading to simplified quantum circuits with improved gate complexity compared to prior nonuniform-quadrature methods.

quant-ph↗

arXiv2025

Quantum algorithm for linear matrix equations

We describe an efficient quantum algorithm for solving the linear matrix equation AX+XB=C, where A, B, and C are given complex matrices and X is unknown. This is known as the Sylvester equation, a fundamental equation with applications in control theory and physics. Our approach constructs the solution matrix X/x in a block-encoding, where x is a rescaling factor needed for normalization. This allows us to obtain certain properties of the entries of X exponentially faster than would be possible from preparing X as a quantum state. The query and gate complexities of the quantum circuit that implements this block-encoding are almost linear in a condition number that depends on A and B, and depend logarithmically in the dimension and inverse error. We show how our quantum circuits can solve BQP-complete problems efficiently, discuss potential applications and extensions of our approach, its connection to Riccati equation, and comment on open problems.

quant-ph↗

arXiv2025

Quantum simulation with sum-of-squares spectral amplification

We present sum-of-squares spectral amplification (SOSSA), a framework for improving quantum simulation relevant to low-energy problems. We show how SOSSA can be applied to problems like energy and phase estimation and provide fast quantum algorithms for these problems that significantly improve over prior art. To illustrate the power of SOSSA in applications, we consider the Sachdev-Ye-Kitaev model, a representative strongly correlated system, and demonstrate asymptotic speedups over generic simulation methods by a factor of the square root of the system size. Our results reinforce those observed in [G.H. Low \textit{et al.}, arXiv:2502.15882 (2025)], where SOSSA was used to achieve state-of-the-art gate costs for phase estimation of real-world quantum chemistry systems.

quant-ph↗

arXiv2025

Quantum Simulation of Boson-Related Hamiltonians: Techniques, Effective Hamiltonian Construction, and Error Analysis

A broad spectrum of physical systems in condensed-matter and high-energy physics, vibrational spectroscopy, and circuit and cavity QED necessitates the incorporation of bosonic degrees of freedom, such as phonons, photons, and gluons, into optimized fermion algorithms for near-future quantum simulations. In particular, when a quantum system is surrounded by an external environment, its basic physics can usually be simplified to a spin or fermionic system interacting with bosonic modes. Nevertheless, troublesome factors such as the magnitude of the bosonic degrees of freedom typically complicate the direct quantum simulation of these interacting models, necessitating the consideration of a comprehensive plan. This strategy should specifically include a suitable fermion/boson-to-qubit mapping scheme to encode sufficiently large yet manageable bosonic modes, and a method for truncating and/or downfolding the Hamiltonian to the defined subspace for performing an approximate but highly accurate simulation, guided by rigorous error analysis. In this pedagogical tutorial review, we aim to provide such an exhaustive strategy, focusing on encoding and simulating certain bosonic-related model Hamiltonians, inclusive of their static properties and time evolutions. Specifically, we emphasize two aspects: (1) the discussion of recently developed quantum algorithms for these interacting models and the construction of effective Hamiltonians, and (2) a detailed analysis regarding a tightened error bound for truncating the bosonic modes for a class of fermion-boson interacting Hamiltonians.

quant-ph↗

arXiv2025

Fast quantum simulation of electronic structure by spectrum amplification

The most advanced techniques using fault-tolerant quantum computers to estimate the ground-state energy of a chemical Hamiltonian involve compression of the Coulomb operator through tensor factorizations, enabling efficient block-encodings of the Hamiltonian. A natural challenge of these methods is the degree to which block-encoding costs can be reduced. We address this challenge through the technique of spectrum amplification, which magnifies the spectrum of the low-energy states of Hamiltonians that can be expressed as sums of squares. Spectrum amplification enables estimating ground-state energies with significantly improved cost scaling in the block encoding normalization factor $Λ$ to just $\sqrt{2ΛE_{\text{gap}}}$, where $E_{\text{gap}} \ll Λ$ is the lowest energy of the sum-of-squares Hamiltonian. To achieve this, we show that sum-of-squares representations of the electronic structure Hamiltonian are efficiently computable by a family of classical simulation techniques that approximate the ground-state energy from below. In order to further optimize, we also develop a novel factorization that provides a trade-off between the two leading Coulomb integral factorization schemes -- namely, double factorization and tensor hypercontraction -- that when combined with spectrum amplification yields a factor of 4 to 195 speedup over the state of the art in ground-state energy estimation for models of Iron-Sulfur complexes and a CO$_{2}$-fixation catalyst.

quant-ph↗

arXiv2024

Classification of electronic structures and state preparation for quantum computation of reaction chemistry

Quantum computation for chemical problems will require the construction of guiding states with sufficient overlap with a target state. Since easily available and initializable mean-field states are characterized by an overlap that is reduced for multi-configurational electronic structures and even vanishes with growing system size, we here investigate the severity of state preparation for reaction chemistry. We emphasize weaknesses in current traditional approaches (even for weakly correlated molecules) and highlight the advantage of quantum phase estimation algorithms. An important result is the introduction of a new classification scheme for electronic structures based on orbital entanglement information. We identify two categories of multi-configurational molecules. Whereas class-1 molecules are dominated by very few determinants and often found in reaction chemistry, class-2 molecules do not allow one to single out a reasonably sized number of important determinants. The latter are particularly hard for traditional approaches and an ultimate target for quantum computation. Some open-shell iron-sulfur clusters belong to class 2. We discuss the role of the molecular orbital basis set and show that true class-2 molecules remain in this class independent of the choice of the orbital basis, with the iron-molybdenum cofactor of nitrogenase being a prototypical example. We stress that class-2 molecules can be build in a systematic fashion from open-shell centers or unsaturated carbon atoms. Our key result is that it will always be possible to initialize a guiding state for chemical reaction chemistry in the ground state based on initial low-cost approximate electronic structure information, which is facilitated by the finite size of the atomistic structures to be considered.

physics.comp-ph↗

arXiv2024

End-to-End Quantum Simulation of a Chemical System

We demonstrate the first end-to-end integration of high-performance computing (HPC), reliable quantum computing, and AI in a case study on catalytic reactions producing chiral molecules. We present a hybrid computation workflow to determine the strongly correlated reaction configurations and estimate, for one such configuration, its active site's ground state energy. We combine 1) the use of HPC tools like AutoRXN and AutoCAS to systematically identify the strongly correlated chemistry within a large chemical space with 2) the use of logical qubits in the quantum computing stage to prepare the quantum ground state of the strongly correlated active site, demonstrating the advantage of logical qubits compared to physical qubits, and 3) the use of optimized quantum measurements of the logical qubits with so-called classical shadows to accurately predict various properties of the ground state including energies. The combination of HPC, reliable quantum computing, and AI in this demonstration serves as a proof of principle of how future hybrid chemistry applications will require integration of large-scale quantum computers with classical computing to be able to provide a measurable quantum advantage.

quant-ph↗

arXiv2024

Classical shadows of fermions with particle number symmetry

We consider classical shadows of fermion wavefunctions with $η$ particles occupying $n$ modes. We prove that all $k$-Reduced Density Matrices (RDMs) may be simultaneously estimated to an average variance of $ε^{2}$ using at most $\binomη{k}\big(1-\frac{η-k}{n}\big)^{k}\frac{1+n}{1+n-k}/ε^{2}$ measurements in random single-particle bases that conserve particle number, and provide an estimator for any $k$-RDM with $\mathcal{O}(k^2η)$ classical complexity. Our sample complexity is a super-exponential improvement over the $\mathcal{O}(\binom{n}{k}\frac{\sqrt{k}}{ε^{2}})$ scaling of prior approaches as $n$ can be arbitrarily larger than $η$, which is common in natural problems. Our method, in the worst-case of half-filling, still provides a factor of $4^{k}$ advantage in sample complexity, and also estimates all $η$-reduced density matrices, applicable to estimating overlaps with all single Slater determinants, with at most $\mathcal{O}(\frac{1}{ε^{2}})$ samples, which is additionally independent of $η$.

quant-ph↗