arXiv subjects

Rolando D. Somma

Publications and source records attributed to Rolando D. Somma.

At least 19 recordsLinked to original sources

arXiv2026

Optimal Lower Bound for Ground-State Energy Estimation with a Guiding State

The guided Hamiltonian problem is the following: given access to the unitary $U=e^{i H}$ for some Hamiltonian $H$, and given access to a unitary that prepares a guiding state promised to have overlap at least $γ>0$ with the ground space of $H$, estimate the ground-state energy of $H$ within additive error $δ> 0$ and success probability at least $1-\varepsilon $, $\varepsilon>0$. How many applications of $U$ and its inverse $U^{-1}$ are necessary and sufficient? An upper bound $O(\log(1/\varepsilon)\log(1/γ)/γδ)$ was known, and was improved to $O(\log(1/\varepsilon)/γδ)$ very recently [JW26]. A matching lower bound was known whenever one of the three parameters $δ,γ,\varepsilon$ was held constant [MdW26]. In this paper we prove the joint lower bound $Ω(\log(1/\varepsilon)/γδ)$ with the tight $\varepsilon$-dependence provided the dimension of $H$ is at least $\log(1/\varepsilon)/γ^2$. Furthermore, we show that this same lower bound (with slightly larger dimension) holds for both the special case in which the ground state is guaranteed to be unique and $H$ has a gap of $δ$ between its first and second eigenvalue; and for ground-state preparation, where $δ$ denotes the spectral gap and $\varepsilon$ now is the approximation error. The lower bounds also apply when the Hamiltonian can be accessed via its block-encoding, and when fractional powers of $U$ are allowed, as in continuous-time Hamiltonian simulation. Lastly, improved upper bounds are known when $H$ is nonnegative and presented as a sum of squares; and our results imply the lower bound $Ω(\log(1/\varepsilon)/γ\sqrtδ)$ for this case.

quant-ph↗

arXiv2026

Efficient quantum algorithm for linear matrix differential equations and applications to open quantum systems

We present an efficient, nearly optimal quantum algorithm for solving linear matrix differential equations, with applications to the simulation of open quantum systems and beyond. For unitary or dissipative dynamics, the algorithm computes an entry of the solution matrix with query complexity $\widetilde{\mathcal{O}}(ν\mathcal{L} t/ε)$, where the constant $ν$ depends on the problem parameters, $\mathcal{L}$ involves a time integral of upper bounds on the norms of evolution operators, and $ε$ is the error. In particular, $ν\mathcal{L}$ is linear in $t$ for unitary dynamics and can be a constant for dissipative dynamics. Our result contrasts prior quantum approaches for differential equations that typically require exponential time for this problem due to the encoding in a quantum state, which can lead to exponentially small amplitudes. We demonstrate the utility of the algorithm through an end-to-end application, namely the simulation of dissipative dynamics for non-interacting fermions, which can be extended to other quantum and classical systems. We compare with classical algorithms and give evidence of polynomial quantum speedups for systems in a lattice, which become more pronounced for systems with long-range interactions and can be shown to be exponential in general. We also provide a lower bound of $Ω(ν\mathcal{L} t/ε)$ for unitary or dissipative dynamics that proves our algorithm is optimal up to logarithmic factors.

quant-ph↗

arXiv2025

Efficient Quantum Hermite Transform

We present a new primitive for quantum algorithms that implements a discrete Hermite transform efficiently, in time that depends logarithmically in both the dimension and the inverse of the allowable error. This transform, which maps basis states to states whose amplitudes are proportional to the Hermite functions, can be interpreted as the Gaussian analogue of the Fourier transform. Our algorithm is based on a method to exponentially fast forward the evolution of the quantum harmonic oscillator, which significantly improves over prior art. We apply this Hermite transform to give examples of provable quantum query advantage in property testing and learning. In particular, we show how to efficiently test the property of being close to a low- degree in the Hermite basis when inputs are sampled from the Gaussian distribution, and how to solve a Gaussian analogue of the Goldreich-Levin learning task efficiently. We also comment on other potential uses of this transform to simulating time dynamics of quantum systems in the continuum.

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

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↗

arXiv2025

Shadow Hamiltonian Simulation

Simulating quantum dynamics is one of the most important applications of quantum computers. Traditional approaches for quantum simulation involve preparing the full evolved state of the system and then measuring some physical quantity. Here, we present a different and novel approach to quantum simulation that uses a compressed quantum state that we call the ``shadow state''. The amplitudes of this shadow state are proportional to the time-dependent expectations of a specific set of operators of interest, and it evolves according to its own Schrödinger equation. This evolution can be simulated on a quantum computer efficiently under broad conditions. Applications of this approach to quantum simulation problems include simulating the dynamics of exponentially large systems of free fermions or free bosons, the latter example recovering a recent algorithm for simulating exponentially many classical harmonic oscillators. These simulations are hard for classical methods and also for traditional quantum approaches, as preparing the full states would require exponential resources. Shadow Hamiltonian simulation can also be extended to simulate expectations of more complex operators such as two-time correlators or Green's functions, and to study the evolution of operators themselves in the Heisenberg picture.

quant-ph↗

arXiv2024

Hamiltonian simulation for low-energy states with optimal time dependence

We consider the task of simulating time evolution under a Hamiltonian $H$ within its low-energy subspace. Assuming access to a block-encoding of $H'=(H-E)/λ$ for some $E \in \mathbb R$, the goal is to implement an $ε$-approximation to $e^{-itH}$ when the initial state is confined to the subspace corresponding to eigenvalues $[-1, -1+Δ/λ]$ of $H'$. We present a quantum algorithm that uses $O(t\sqrt{λΓ} + \sqrt{λ/Γ}\log(1/ε))$ queries to the block-encoding for any $Γ$ such that $Δ\leq Γ\leq λ$. When $\log(1/ε) = o(tλ)$ and $Δ/λ= o(1)$, this result improves over generic methods with query complexity $Ω(tλ)$. Our quantum algorithm leverages spectral gap amplification and the quantum singular value transform. Using standard access models for $H$, we show that the ability to efficiently block-encode $H'$ is equivalent to $H$ being what we refer to as a "gap-amplifiable" Hamiltonian. This includes physically relevant examples such as frustration-free systems, and it encompasses all previously considered settings of low-energy simulation algorithms. We also provide lower bounds for low-energy simulation. In the worst case, we show that the low-energy condition cannot be used to improve the runtime of Hamiltonian simulation. For gap-amplifiable Hamiltonians, we prove that our algorithm is tight in the query model with respect to $t$, $Δ$, and $λ$. In the practically relevant regime where $\log (1/ε) = o(tΔ)$ and $Δ/λ= o(1)$, we also prove a matching lower bound in gate complexity (up to log factors). To establish the query lower bounds, we consider $\mathrm{PARITY}\circ\mathrm{OR}$ and degree bounds on trigonometric polynomials. To establish the lower bound on gate complexity, we use a circuit-to-Hamiltonian reduction acting on a low-energy state.

quant-ph↗

arXiv2024

Kernel Polynomial Method for Linear Spin Wave Theory

Calculating dynamical spin correlations is essential for matching model magnetic exchange Hamiltonians to momentum-resolved spectroscopic measurements. A major numerical bottleneck is the diagonalization of the dynamical matrix, especially in systems with large magnetic unit cells, such as those with incommensurate magnetic structures or quenched disorder. In this paper, we demonstrate an efficient scheme based on the kernel polynomial method for calculating dynamical correlations of relevance to inelastic neutron scattering experiments. This method reduces the scaling of numerical cost from cubic to linear in the magnetic unit cell size.

cond-mat.str-el↗

arXiv2023

Exponential quantum speedup in simulating coupled classical oscillators

We present a quantum algorithm for simulating the classical dynamics of $2^n$ coupled oscillators (e.g., $2^n$ masses coupled by springs). Our approach leverages a mapping between the Schrödinger equation and Newton's equation for harmonic potentials such that the amplitudes of the evolved quantum state encode the momenta and displacements of the classical oscillators. When individual masses and spring constants can be efficiently queried, and when the initial state can be efficiently prepared, the complexity of our quantum algorithm is polynomial in $n$, almost linear in the evolution time, and sublinear in the sparsity. As an example application, we apply our quantum algorithm to efficiently estimate the kinetic energy of an oscillator at any time. We show that any classical algorithm solving this same problem is inefficient and must make $2^{Ω(n)}$ queries to the oracle and, when the oracles are instantiated by efficient quantum circuits, the problem is BQP-complete. Thus, our approach solves a potentially practical application with an exponential speedup over classical computers. Finally, we show that under similar conditions our approach can efficiently simulate more general classical harmonic systems with $2^n$ modes.

quant-ph↗

arXiv2022

Quantum algorithms from fluctuation theorems: Thermal-state preparation

Fluctuation theorems provide a correspondence between properties of quantum systems in thermal equilibrium and a work distribution arising in a non-equilibrium process that connects two quantum systems with Hamiltonians $H_0$ and $H_1=H_0+V$. Building upon these theorems, we present a quantum algorithm to prepare a purification of the thermal state of $H_1$ at inverse temperature $β\ge 0$ starting from a purification of the thermal state of $H_0$. The complexity of the quantum algorithm, given by the number of uses of certain unitaries, is $\tilde {\cal O}(e^{β(Δ\! A- w_l)/2})$, where $Δ\! A$ is the free-energy difference between $H_1$ and $H_0,$ and $w_l$ is a work cutoff that depends on the properties of the work distribution and the approximation error $ε>0$. If the non-equilibrium process is trivial, this complexity is exponential in $β\|V\|$, where $\|V\|$ is the spectral norm of $V$. This represents a significant improvement of prior quantum algorithms that have complexity exponential in $β\|H_1\|$ in the regime where $\|V\|\ll \|H_1\|$. The dependence of the complexity in $ε$ varies according to the structure of the quantum systems. It can be exponential in $1/ε$ in general, but we show it to be sublinear in $1/ε$ if $H_0$ and $H_1$ commute, or polynomial in $1/ε$ if $H_0$ and $H_1$ are local spin systems. The possibility of applying a unitary that drives the system out of equilibrium allows one to increase the value of $w_l$ and improve the complexity even further. To this end, we analyze the complexity for preparing the thermal state of the transverse field Ising model using different non-equilibrium unitary processes and see significant complexity improvements.

quant-ph↗

arXiv2022

Tensor networks for High Energy Physics: contribution to Snowmass 2021

Tensor network methods are becoming increasingly important for high-energy physics, condensed matter physics and quantum information science (QIS). We discuss the impact of tensor network methods on lattice field theory, quantum gravity and QIS in the context of High Energy Physics (HEP). These tools will target calculations for strongly interacting systems that are made difficult by sign problems when conventional Monte Carlo and other importance sampling methods are used. Further development of methods and software will be needed to make a significant impact in HEP. We discuss the roadmap to perform quantum chromodynamics (QCD) related calculations in the coming years. The research is labor intensive and requires state of the art computational science and computer science input for its development and validation. We briefly discuss the overlap with other science domains and industry.

hep-lat↗

arXiv2021

Fast-forwarding quantum evolution

We investigate the problem of fast-forwarding quantum evolution, whereby the dynamics of certain quantum systems can be simulated with gate complexity that is sublinear in the evolution time. We provide a definition of fast-forwarding that considers the model of quantum computation, the Hamiltonians that induce the evolution, and the properties of the initial states. Our definition accounts for any asymptotic complexity improvement of the general case and we use it to demonstrate fast-forwarding in several quantum systems. In particular, we show that some local spin systems whose Hamiltonians can be taken into block diagonal form using an efficient quantum circuit, such as those that are permutation-invariant, can be exponentially fast-forwarded. We also show that certain classes of positive semidefinite local spin systems, also known as frustration-free, can be polynomially fast-forwarded, provided the initial state is supported on a subspace of sufficiently low energies. Last, we show that all quadratic fermionic systems and number-conserving quadratic bosonic systems can be exponentially fast-forwarded in a model where quantum gates are exponentials of specific fermionic or bosonic operators, respectively. Our results extend the classes of physical Hamiltonians that were previously known to be fast-forwarded, while not necessarily requiring methods that diagonalize the Hamiltonians efficiently. We further develop a connection between fast-forwarding and precise energy measurements that also accounts for polynomial improvements.

quant-ph↗

arXiv2021

Hamiltonian simulation in the low-energy subspace

We study the problem of simulating the dynamics of spin systems when the initial state is supported on a subspace of low energy of a Hamiltonian $H$. This is a central problem in physics with vast applications in many-body systems and beyond, where the interesting physics takes place in the low-energy sector. We analyze error bounds induced by product formulas that approximate the evolution operator and show that these bounds depend on an effective low-energy norm of $H$. We find improvements over the best previous complexities of product formulas that apply to the general case, and these improvements are more significant for long evolution times that scale with the system size and/or small approximation errors. To obtain these improvements, we prove exponentially decaying upper bounds on the leakage to high-energy subspaces due to the product formula. Our results provide a path to a systematic study of Hamiltonian simulation at low energies, which will be required to push quantum simulation closer to reality.

quant-ph↗

arXiv2021

Complexity of quantum state verification in the quantum linear systems problem

We analyze the complexity of quantum state verification in the context of solving systems of linear equations of the form $A \vec x = \vec b$. We show that any quantum operation that verifies whether a given quantum state is within a constant distance from the solution of the quantum linear systems problem requires $q=Ω(κ)$ uses of a unitary that prepares a quantum state $\left| b \right>$, proportional to $\vec b$, and its inverse in the worst case. Here, $κ$ is the condition number of the matrix $A$. For typical instances, we show that $q=Ω(\sqrt κ)$ with high probability. These lower bounds are almost achieved if quantum state verification is performed using known quantum algorithms for the quantum linear systems problem. We also analyze the number of copies of $\left| b \right>$ required by verification procedures of the prepare and measure type. In this case, the lower bounds are quadratically worse, being $Ω(κ^2)$ in the worst case and $Ω(κ)$ in typical instances with high probability. We discuss the implications of our results to known variational and related approaches to this problem, where state preparation, gate, and measurement errors will need to decrease rapidly with $κ$ for worst-case and typical instances if error correction is not used, and present some open problems.

quant-ph↗

arXiv2020

Computing partition functions in the one clean qubit model

We present a method to approximate partition functions of quantum systems using mixed-state quantum computation. For positive semi-definite Hamiltonians, our method has expected running-time that is almost linear in $(M/(ε_{\rm rel}\mathcal{Z} ))^2$, where $M$ is the dimension of the quantum system, $\mathcal{Z}$ is the partition function, and $ε_{\rm rel}$ is the relative precision. It is based on approximations of the exponential operator as linear combinations of certain operators related to block-encoding of Hamiltonians or Hamiltonian evolutions. The trace of each operator is estimated using a standard algorithm in the one clean qubit model. For large values of $\mathcal{Z}$, our method may run faster than exact classical methods, whose complexities are polynomial in $M$. We also prove that a version of the partition function estimation problem within additive error is complete for the so-called DQC1 complexity class, suggesting that our method provides a super-polynomial speedup for certain parameter values. To attain a desired relative precision, we develop a classical procedure based on a sequence of approximations within predetermined additive errors that may be of independent interest.

quant-ph↗

arXiv2020

Quantum eigenvalue estimation via time series analysis

We present an efficient method for estimating the eigenvalues of a Hamiltonian $H$ from the expectation values of the evolution operator for various times. For a given quantum state $ρ$, our method outputs a list of eigenvalue estimates and approximate probabilities. Each probability depends on the support of $ρ$ in those eigenstates of $H$ associated with eigenvalues within an arbitrarily small range. The complexity of our method is polynomial in the inverse of a given precision parameter $ε$, which is the gap between eigenvalue estimates. Unlike the well-known quantum phase estimation algorithm that uses the quantum Fourier transform, our method does not require large ancillary systems, large sequences of controlled operations, or preserving coherence between experiments, and is therefore more attractive for near-term applications. The output of our method can be used to compute spectral properties of $H$ and other expectation values efficiently, within additive error proportional to $ε$.

quant-ph↗