arXiv subjects

Nathan Wiebe

Publications and source records attributed to Nathan Wiebe.

At least 37 records · Page 2Linked to original sources

arXiv2025

First-Quantized Quantum Simulation of Non-Relativistic QED with Emergent Topologically Protected Coulomb Interactions

We provide a simulation algorithm that properly addresses light matter interaction between non-relativistic first-quantized charged particles and quantum electromagnetic fields. Unlike previous work, our Hamiltonian does not include an explicit Coulomb interaction between particles. Rather, the Coulomb interaction emerges from the imposition of Gauss' law as a constraint upon the system in an appropriate non-relativistic limit. Furthermore, a form of topological protection emerges in our formalism, analogous to that of the Toric code Hamiltonian. This mechanism prevents simulation-induced electric field errors that can be contracted to a point from causing any deviations from Coulomb's law in the non-relativistic limit and any error that forms a non-contractable loop is energetically dissallowed in the limit of large volume. We find that, under appropriate continuity assumptions, the number of non-Clifford gates required by our algorithm scales in the thermodynamic limit as $\widetilde{O}(N^{2/3}η^{4/3} t \log^5(1/ε))$ for $η$ particles, $N$ spatial grid points, simulation time $t$ and error tolerance $ε$. In comparison, the more specific problem of simulating the Coulomb interaction scales as $\widetilde{O}(N^{1/3} η^{8/3} t \log^2(1/ε))$. This suggests that if $N \in \tilde{o}(η^4)$ that our non-relativistic electrodynamic simulation method could provide a computational advantage for electronic structure problems in the thermodynamic limit under appropriate continuity assumptions as it obviates the need to compute the $O(η^2)$ pairwise interactions in the Coulomb Hamiltonian.

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arXiv2025

Hybrid Oscillator-Qubit Quantum Processors: Instruction Set Architectures, Abstract Machine Models, and Applications

Quantum computing with discrete variable (DV, qubit) hardware is approaching the large scales necessary for computations beyond the reach of classical computers. However, important use cases such as quantum simulations of physical models containing bosonic modes, and quantum error correction are challenging for DV-only systems. Separately, hardware containing native continuous-variable (CV, oscillator) systems has received attention as an alternative approach, yet the universal control of such systems is non-trivial. In this work, we show that hybrid CV-DV hardware offers a great advantage in meeting these challenges, offering a powerful computational paradigm that inherits the strengths of both DV and CV processors. We provide a pedagogical introduction to CV-DV systems and the multiple abstraction layers needed to produce a full software stack connecting applications to hardware. We present a variety of new hybrid CV-DV compilation techniques, algorithms, and applications, including the extension of quantum signal processing concepts to CV-DV systems and strategies to simulate systems of interacting spins, fermions, and bosons. To facilitate the development of hybrid CV-DV processor systems, we introduce formal Abstract Machine Models and Instruction Set Architectures -- essential abstractions that enable developers to formulate applications, compile algorithms, and explore the potential of current and future hardware for realizing fault-tolerant circuits, modules, and processors. Hybrid CV-DV quantum computations are beginning to be performed in superconducting, trapped ion, and neutral atom platforms, and large-scale experiments are set to be demonstrated in the near future. We present a timely and comprehensive guide to this relatively unexplored yet promising approach to quantum computation and providing an architectural backbone to guide future development.

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arXiv2025

Calibration of Quantum Devices via Robust Statistical Methods

Bayesian inference is a widely used technique for real-time characterization of quantum systems. It excels in experimental characterization in the low data regime, and when the measurements have degrees of freedom. A decisive factor for its performance is the numerical representation of the Bayesian probability distributions. In this work, we explore advanced statistical methods for this purpose, and numerically analyze their performance against the state-of-the-art in quantum parameter learning. In particular, we consider sequential importance resampling, tempered likelihood estimation, Markov Chain Monte Carlo, random walk Metropolis (RWM), Hamiltonian Monte Carlo (HMC) and variants (stochastic gradients with and without friction, energy conserving subsampling), block pseudo-marginal Metropolis-Hastings with subsampling, hybrid HMC-RWM approaches, and Gaussian rejection filtering. We demonstrate advantages of these approaches over existing ones, namely robustness under multi-modality and high dimensionality. We apply these algorithms to the calibration of superconducting qubits from IBMQ, surpassing the standard quantum limit and achieving better results than Qiskit's default tools. In Hahn echo and Ramsey experiments, we reduce the uncertainty by factors of 10 and 3 respectively, without increasing the number of measurements; conversely, we match the performance of Qiskit's methods while using up to to 99.5% less experimental data. We additionally investigate the roles of adaptivity, dataset ordering and heuristics in quantum characterization. Our findings have applications in challenging quantum characterization tasks, namely learning the dynamics of open quantum systems.

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arXiv2025

Application-level Benchmarking of Quantum Computers using Nonlocal Game Strategies

In a nonlocal game, two noncommunicating players cooperate to convince a referee that they possess a strategy that does not violate the rules of the game. Quantum strategies allow players to optimally win some games by performing joint measurements on a shared entangled state, but computing these strategies can be challenging. We present a variational quantum algorithm to compute quantum strategies for nonlocal games by encoding the rules of a nonlocal game into a Hamiltonian. We show how this algorithm can generate a short-depth optimal quantum strategy for a graph coloring game with a quantum advantage. This quantum strategy is then evaluated on fourteen different quantum hardware platforms to demonstrate its utility as a benchmark. Finally, we discuss potential sources of errors that can explain the observed decreased performance of the executed task and derive an expression for the number of samples required to accurately estimate the win rate in the presence of noise.

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arXiv2025

Arbitrary Boundary Conditions and Constraints in Quantum Algorithms for Differential Equations via Penalty Projections

Complicated boundary conditions are essential to accurately describe phenomena arising in nature and engineering. Recently, the investigation of a potential speedup through quantum algorithms in simulating the governing ordinary and partial differential equations of such phenomena has gained increasing attention. We design an efficient quantum algorithms for solving differential equations with arbitrary boundary conditions. Specifically, we propose an approach to enforce arbitrary boundary conditions and constraints through adding a penalty projection to the governing equations. Assuming a fast-forwardable representation of the projection to ensure an efficient interaction picture imulation, the cost of to enforce the constraints is at most $O(\logλ)$ in the strength of the penalty $λ$ in the gate complexity; in the worst case, this goes as $O([\|v(0)\|^2\|A_0\| + \|b\|_{L^1[0;t]}^2)]t^2/\varepsilon)$, for precision $\varepsilon$ and a dynamical system $\frac{\rm d}{{\rm d}t} v(t) = A_0(t) v(t) + b(t)$ with negative semidefinite $A_0(t)$ of size $n^d\times n^d$. E.g., for the heat equation, this leads to a gate complexity overhead of $\widetilde O(d\log n + \log t)$. We show constraint error bounds for the penalty approach and provide validating numerical experiments, and estimate the circuit complexity using the Linear Combination of Hamiltonian Simulation.

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arXiv2025

Productionizing Quantum Mass Production

For many practical applications of quantum computing, the most costly steps involve coherently accessing classical data. We help address this challenge by applying mass production techniques, which can reduce the cost of applying an operation multiple times in parallel. We combine these techniques with modern approaches for classical data loading based on "quantum read-only memory" (QROM). We find that we can polynomially reduce the total number of gates required for data loading, but we find no advantage in cost models that only count the number of non-Clifford gates. Furthermore, for realistic cost models and problem sizes, we find that it is possible to reduce the cost of parallel data loading by an order of magnitude or more. We present several applications of quantum mass production, including a scheme that uses parallel phase estimation to asymptotically reduce the gate complexity of state-of-the-art algorithms for estimating eigenvalues of the quantum chemical Hamiltonian, including both Clifford and non-Clifford gates, from $\widetilde{\mathcal{O}}\left(N_{orb}^2\right)$ to $\widetilde{\mathcal{O}}\left(N_{orb}^{\log_2 3}\right)$, where $N_{orb}$ denotes the number of orbitals. We also show that mass production can be used to reduce the cost of serial calls to the same data loading oracle by precomputing several copies of a novel QROM resource state.

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arXiv2025

Simulating Time Dependent and Nonlinear Classical Oscillators through Nonlinear Schrödingerization

We present quantum algorithms for simulating the dynamics of a broad class of classical oscillator systems containing $2^n$ coupled oscillators (Eg: $2^n$ masses coupled by springs), including those with time-dependent forces, time-varying stiffness matrices, and weak nonlinear interactions. This generalization of the Harmonic oscillator simulation algorithm is achieved through an approach that we call ``Nonlinear Schrödingerization'', which involves reduction of the dynamical system to a nonlinear Schrödinger equation and then reduced to a time-independent Schrodinger Equation through perturbative techniques. The linearization of the equation is performed using an approach that allows the dynamics of a nonlinear Schrödinger equation to be approximated as a linear Schrödinger equation in a higher dimensional space. This allows Hamiltonian Simulation algorithms to be applied to simulate the dynamics of resulting system. When the properties of the classical dynamical systems can be efficiently queried, and when the initial state can be efficiently prepared, the complexity of our quantum algorithm is polynomial in $n$, and almost linear in evolution time for most dynamical systems. Our work extends the applicability of quantum algorithms to simulate the dynamics of non-conservative and nonlinear classical systems, addressing key limitations in previous approaches.

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arXiv2025

Interpreting Multipartite Entanglement through Topological Summaries

The study of multipartite entanglement is much less developed than the bipartite scenario. Recently, a string of results have proposed using tools from Topological Data Analysis (TDA) to attach topological quantities to multipartite states. However, these quantities are not directly connected to concrete information processing tasks making their interpretations vague. We take the first steps in connecting these abstract topological quantities to operational interpretations of entanglement in two scenarios. The first is we provide a bound on the Integrated Euler Characteristic defined by Hamilton and Leditzky via an average distillable entanglement, which we develop as a generalization of the Meyer-Wallach entanglement measure studied by A. J. Scott in 2004. This allows us to connect the distance of an error correcting code to the Integrated Euler Characteristic. The second is we provide a characterization of a class of graph states containing the GHZ state via the birth and death times of the connected components and 1-dimensional cycles of the entanglement complex. In other words, the entanglement distance behavior of the first Betti number $β_1(\varepsilon)$ allows us to determine if a state is locally equivalent to a GHZ state, potentially providing new verification schemes for highly entangled states.

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arXiv2025

Quantum and classical algorithms for nonlinear unitary dynamics

Quantum algorithms for Hamiltonian simulation and linear differential equations more generally have provided promising exponential speed-ups over classical computers on a set of problems with high real-world interest. However, extending this to a nonlinear problem has proven challenging, with exponential lower bounds having been demonstrated for the time scaling. We provide a quantum algorithm matching these bounds. Specifically, we find that for a non-linear differential equation of the form $\frac{d|u\rangle}{dt} = A|u\rangle + B|u\rangle^{\otimes2}$ for evolution of time $T$, error tolerance $ε$ and $c$ dependent on the strength of the nonlinearity, the number of queries to the differential operators that approaches the scaling of the quantum lower bound of $e^{o(T\|B\|)}$ queries in the limit of strong non-linearity. Finally, we introduce a classical algorithm based on the Euler method allowing comparably scaling to the quantum algorithm in a restricted case, as well as a randomized classical algorithm based on path integration that acts as a true analogue to the quantum algorithm in that it scales comparably to the quantum algorithm in cases where sign problems are absent.

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arXiv2025

Dividing and Conquering the Van Vleck Catastrophe

The quantum-computational cost of determining ground state energies through quantum phase estimation depends on the overlap between an easily preparable initial state and the targeted ground state. The Van Vleck orthogonality catastrophe has frequently been invoked to suggest that quantum computers may not be able to efficiently prepare ground states of large systems because the overlap with the initial state tends to decrease exponentially with the system size, even for non-interacting systems. We show that this intuition is not necessarily true. Specifically, we introduce a divide-and-conquer strategy that repeatedly uses phase estimation to merge ground states of increasingly larger subsystems. We provide rigorous bounds for this approach and show that if the minimum success probability of each merge is lower bounded by a constant, then the query complexity of preparing the ground state of $N$ interacting systems is in $O(N^{\log\log(N)} {\rm poly}(N))$, which is quasi-polynomial in $N$, in contrast to the exponential scaling anticipated by the Van Vleck catastrophe. We also discuss sufficient conditions on the Hamiltonian that ensure a quasi-polynomial running time.

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arXiv2025

Quantum Simulation of Lindbladian Dynamics via Repeated Interactions

The Lindblad equation generalizes the Schrödinger equation to quantum systems that undergo dissipative dynamics. The quantum simulation of Lindbladian dynamics is therefore non-unitary, preventing a naive application of state-of-the-art quantum algorithms. Here, we make use of an approximate correspondence between Lindbladian dynamics and evolution based on Repeated Interaction (RI) CPTP maps to write down a Hamiltonian formulation of the Lindblad dynamics and derive a rigorous error bound on the master equation. Specifically, we show that the number of interactions needed to simulate the Liouvillian $e^{t\mathcal{L}}$ within error $ε$ scales in a weak coupling limit as $ν\in O(t^2\|\mathcal{L}\|_{1\rightarrow 1}^2/ε)$. This is significant because the error in the Lindbladian approximation to the dynamics is not explicitly bounded in existing quantum algorithms for open system simulations. We then provide quantum algorithms to simulate RI maps using an iterative Qubitization approach and Trotter-Suzuki formulas and specifically show that for iterative Qubitization the number of operations needed to simulate the dynamics (for a fixed value of $ν$) scales in a weak coupling limit as $O(α_0 t + ν\log(1/ε)/\log\log(1/ε))$ where $α_0$ is the coefficient $1$-norm for the system and bath Hamiltonians. This scaling would appear to be optimal if the complexity of $ν$ is not considered, which underscores the importance of considering the error in the Liouvillian that we reveal in this work.

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arXiv2025

Hamiltonian Learning via Shadow Tomography of Pseudo-Choi States

We introduce a new approach to learn Hamiltonians through a resource that we call the pseudo-Choi state, which encodes the Hamiltonian in a state using a procedure that is analogous to the Choi-Jamiolkowski isomorphism. We provide an efficient method for generating these pseudo-Choi states by querying a time evolution unitary of the form $e^{-iHt}$ and its inverse, and show that for a Hamiltonian with $M$ terms the Hamiltonian coefficients can be estimated via classical shadow tomography within error $ε$ in the $2$-norm using $\widetilde{O}\left(\frac{M}{t^2ε^2}\right)$ queries to the state preparation protocol, where $t \le \frac{1}{2\left\lVert H \right\rVert}$. We further show an alternative approach that eschews classical shadow tomography in favor of quantum mean estimation that reduces this cost (at the price of many more qubits) to $\widetilde{O}\left(\frac{M}{tε}\right)$. Additionally, we show that in the case where one does not have access to the state preparation protocol, the Hamiltonian can be learned using $\widetilde{O}\left(\frac{α^4M}{ε^2}\right)$ copies of the pseudo-Choi state. The constant $α$ depends on the norm of the Hamiltonian, and the scaling in terms of $α$ can be improved quadratically if using pseudo-Choi states of the normalized Hamiltonian. Finally, we show that our learning process is robust to errors in the resource states and to errors in the Hamiltonian class. Specifically, we show that if the true Hamiltonian contains more terms than we believe are present in the reconstruction, then our methods give an indication that there are Hamiltonian terms that have not been identified and will still accurately estimate the known terms in the Hamiltonian.

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arXiv2025

The Thermodynamic Cost of Ignorance: Thermal State Preparation with One Ancilla Qubit

In this work we investigate a model of thermalization wherein a single ancillary qubit randomly interacts with the system to be thermalized. This not only sheds light on the emergence of Gibbs states in nature, but also provides a routine for preparing arbitrary thermal states on a digital quantum computer. For desired $β$ and random interaction $G$ the routine boils down to time independent Hamiltonian simulation and is represented by the channel $Φ: ρ\mapsto \mathbb{E}_G {\rm Tr}_{\rm Env} \left[ e^{-i(H + αG)t} \left(ρ\otimes \frac{e^{-βH_E}}{\mathcal{Z}}\right) e^{i (H + αG)t} \right]$. We rigorously prove that these dynamics reduce to a Markov chain process in the weak-coupling regime with the thermal state as the approximate fixed point. We upper bound the total simulation time required in terms of the Markov chain spectral gap $λ_\star$, which we compute exactly in the ground state limit. These results are independent of any eigenvalue knowledge of the system, but we are further able to show that with knowledge of eigenvalue differences $λ_S(i) - λ_S(j)$, then the total simulation time is dramatically reduced. The ratio of the complete ignorance simulation cost to the perfect knowledge simulation cost scales as $\widetilde{O} \left({\frac{\|{H_S}\|^7}{δ_{\rm min}^7 ε^{3.5} λ_\star(β)^{3.5}}}\right)$, where $δ_{\min}$ is related to the eigenvalue differences of the system. Additionally, we provide more specific results for single qubit and harmonic oscillator systems as well as numeric experiments with hydrogen chains. In addition to the algorithmic merits, these results can be viewed as broad extensions of the Repeated Interactions model to generic Hamiltonians with unknown interactions, giving a complete picture of the thermalization process for quantum systems.

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arXiv2025

Leveraging Hamiltonian Simulation Techniques to Compile Operations on Bosonic Devices

Circuit QED enables the combined use of qubits and oscillator modes. Despite a variety of available gate sets, many hybrid qubit-boson (i.e., oscillator) operations are realizable only through optimal control theory (OCT) which is oftentimes intractable and uninterpretable. We introduce an analytic approach with rigorously proven error bounds for realizing specific classes of operations via two matrix product formulas commonly used in Hamiltonian simulation, the Lie--Trotter and Baker--Campbell--Hausdorff product formulas. We show how this technique can be used to realize a number of operations of interest, including polynomials of annihilation and creation operators, i.e., $a^p {a^\dagger}^q$ for integer $p, q$. We show examples of this paradigm including: obtaining universal control within a subspace of the entire Fock space of an oscillator, state preparation of a fixed photon number in the cavity, simulation of the Jaynes--Cummings Hamiltonian, simulation of the Hong-Ou-Mandel effect and more. This work demonstrates how techniques from Hamiltonian simulation can be applied to better control hybrid boson-qubit devices.

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arXiv2024

Efficient Quantum Simulation Algorithms in the Path Integral Formulation

We provide a new paradigm for quantum simulation that is based on path integration that allows quantum speedups to be observed for problems that are more naturally expressed using the path integral formalism rather than the conventional sparse Hamiltonian formalism. We provide two novel quantum algorithms based on Hamiltonian versions of the path integral formulation and another for Lagrangians of the form $\frac{m}{2}\dot{x}^2 - V(x)$. This Lagrangian path integral algorithm is based on a new rigorous derivation of a discrete version of the Lagrangian path integral. Our first Hamiltonian path integral method breaks up the paths into short timesteps. It is efficient under appropriate sparsity assumptions and requires a number of queries to oracles that give the eigenvalues and overlaps between the eigenvectors of the Hamiltonian terms that scales as $t^{o(1)}/ε^{o(1)}$ for simulation time $t$ and error $ε$. The second approach uses long-time path integrals for near-adiabatic systems and has query complexity that scales as $O(1/\sqrtε)$ if the energy eigenvalue gaps and simulation time is sufficiently long. Finally, we show that our Lagrangian simulation algorithm requires a number of queries to an oracle that computes the discrete Lagrangian that scales for a system with $η$ particles in $D+1$ dimensions, in the continuum limit, as $\widetilde{O}(ηD t^2/ε)$ if $V(x)$ is bounded and finite and the wave function obeys appropriate position and momentum cutoffs. This shows that Lagrangian dynamics can be efficiently simulated on quantum computers and opens up the possibility for quantum field theories for which the Hamiltonian is unknown to be efficiently simulated on quantum computers.

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arXiv2024

Hybrid Oscillator-Qubit Quantum Processors: Simulating Fermions, Bosons, and Gauge Fields

We develop a hybrid oscillator-qubit processor framework for quantum simulation of strongly correlated fermions and bosons that avoids the boson-to-qubit mapping overhead encountered in qubit hardware. This framework gives exact decompositions of particle interactions such as density-density terms and gauge-invariant hopping, as well as approximate methods based on the Baker-Campbell Hausdorff formulas including the magnetic field term for the $U(1)$ quantum link model in $(2+1)$D. We use this framework to show how to simulate dynamics using Trotterisation, perform ancilla-free partial error detection using Gauss's law, measure non-local observables, estimate ground state energies using a oscillator-qubit variational quantum eigensolver as well as quantum signal processing, and we numerically study the influence of hardware errors in circuit QED experiments. To show the advantages over all-qubit hardware, we perform an end-to-end comparison of the gate complexity for the gauge-invariant hopping term and find an improvement of the asymptotic scaling with the boson number cutoff $S$ from $\mathcal{O}(\log(S)^2)$ to $\mathcal{O}(1)$ in our framework as well as, for bosonic matter, a constant factor improvement of better than $10^4$. We also find an improvement from $\mathcal{O}(\log(S))$ to $\mathcal{O}(1)$ for the $U(1)$ magnetic field term. While our work focusses on an implementation in superconducting hardware, our framework can also be used in trapped ion, and neutral atom hardware. This work establishes digital quantum simulation with hybrid oscillator-qubit hardware as a viable and advantageous method for the study of qubit-boson models in materials science, chemistry, and high-energy physics.

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arXiv2024

Multivariable QSP and Bosonic Quantum Simulation using Iterated Quantum Signal Processing

We provide in this work a form of Modular Quantum Signal Processing that we call iterated quantum signal processing. This method recursively applies quantum signal processing to the outputs of other quantum signal processing steps, allowing polynomials to be easily achieved that would otherwise be difficult to find analytically. We specifically show by using a squaring quantum signal processing routine, that multiplication of phase angles can be approximated and in turn that any bounded degree multi-variate polynomial function of a set of phase angles can be implemented using traditional QSP ideas. We then discuss how these ideas can be used to construct phase functions relevant for quantum simulation such as the Coulomb potential and also discuss how to use these ideas to obviate the need for reversible arithmetic to compute square-root functions needed for simulations of bosonic Hamiltonians.

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arXiv2024

Ground State Preparation via Dynamical Cooling

Quantum algorithms for probing ground-state properties of quantum systems require good initial states. Projection-based methods such as eigenvalue filtering rely on inputs that have a significant overlap with the low-energy subspace, which can be challenging for large, strongly-correlated systems. This issue has motivated the study of physically-inspired dynamical approaches such as thermodynamic cooling. In this work, we introduce a ground-state preparation algorithm based on the simulation of quantum dynamics. Our main insight is to transform the Hamiltonian by a shifted sign function via quantum signal processing, effectively mapping eigenvalues into positive and negative subspaces separated by a large gap. This automatically ensures that all states within each subspace conserve energy with respect to the transformed Hamiltonian. Subsequent time-evolution with a perturbed Hamiltonian induces transitions to lower-energy states while preventing unwanted jumps to higher energy states. The approach does not rely on a priori knowledge of energy gaps and requires no additional qubits to model a bath. Furthermore, it makes $\tilde{\mathcal{O}}(d^{\,3/2}/ε)$ queries to the time-evolution operator of the system and $\tilde{\mathcal{O}}(d^{\,3/2})$ queries to a block-encoding of the perturbation, for $d$ cooling steps and an $ε$-accurate energy resolution. Our results provide a framework for combining quantum signal processing and Hamiltonian simulation to design heuristic quantum algorithms for ground-state preparation.

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