arXiv subjects

N. Anders Petersson

Publications and source records attributed to N. Anders Petersson.

At least 19 recordsLinked to original sources

arXiv2026

Recovering Complex Unitary Eigenspaces from Real-Valued Embeddings

We consider the problem of recovering a unitary eigendecomposition of a complex unitary matrix from that of its embedded real-valued formulation. Such formulations arise naturally in scientific computing workflows that employ real-arithmetic solvers by representing complex matrices in term of their real and imaginary parts. While the reconstruction is trivial when the spectrum of the real-valued embedding is simple, degenerate and/or complex conjugated eigenvalues introduce ambiguities because each eigenspace may include contributions from both the unitary matrix and its complex conjugate. We prove that this ambiguity can always be resolved by applying a structured projection to the eigenspaces of the real-valued embedding, followed by a rank-revealing orthonormalization. The resulting procedure recovers the eigenvalues and an unitary eigenbasis for the original unitary matrix, with correct multiplicities of degenerate eigenvalues.

math.NA↗

arXiv2026

Dynamical Simulations of Schrödinger's Equation via Rank-Adaptive Tensor Decompositions

We study low-rank tensor methods for the numerical solution of Schrödinger's equation with time-independent and explicitly time-dependent Hamiltonians, motivated by large-scale simulations of many-body quantum systems and quantum computing devices subject to time-dependent control pulses. We outline the recent application of the "basis update and Galerkin" (BUG) method for tensor trains, and describe the established TDVP and TDVP-2 algorithms based on the time-dependent variational principle. For comparison, we also consider the BUG method in the Tucker format. All these approaches enable memory efficient representations of partially entangled quantum states and thereby mitigate the exponential cost of conventional state-vector formulations. The rank-adaptivity relies on the truncated singular value decomposition, in which the rank of a matrix is reduced by setting its smallest singular values to zero, based on a threshold parameter that controls the truncation error. Numerical experiments on representative time-independent and time-dependent Hamiltonian models quantify the tradeoff between accuracy and compression across methods, with particular attention to the interplay between the time-step and the truncation threshold, and how the computational effort scales with the number of sub-systems in the quantum system.

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arXiv2025

Non-degenerate Marginal-Likelihood Calibration with Application to Quantum Characterization

We propose a marginal likelihood strategy within the Kennedy-O'Hagan (KOH) Bayesian framework, where a Gaussian process (GP) models the discrepancy between a physical system and its simulator. Our approach introduces a novel marginalized likelihood by integrating out the degenerate eigenspace of the covariance matrix, rather than approximating the original likelihood. Unlike approximation methods that compromise accuracy for computational efficiency, our method defines an exact likelihood -- distinct from the original but preserving all relevant information. This formulation achieves computational efficiency and stability, even for large datasets where the covariance matrix nears degeneracy. Applied to the characterization of a superconducting quantum device at Lawrence Livermore National Laboratory, the approach enhances the predictive accuracy of the Lindblad master equations for modeling Ramsey measurement data by effectively quantifying uncertainties consistent with the quantum data.

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arXiv2024

A time-parallel multiple-shooting method for large-scale quantum optimal control

Quantum optimal control plays a crucial role in quantum computing by providing the interface between compiler and hardware. Solving the optimal control problem is particularly challenging for multi-qubit gates, due to the exponential growth in computational complexity with the system's dimensionality and the deterioration of optimization convergence. To ameliorate the computational complexity of time-integration, this paper introduces a multiple-shooting approach in which the time domain is divided into multiple windows and the intermediate states at window boundaries are treated as additional optimization variables. This enables parallel computation of state evolution across time-windows, significantly accelerating objective function and gradient evaluations. Since the initial state matrix in each window is only guaranteed to be unitary upon convergence of the optimization algorithm, the conventional gate trace infidelity is replaced by a generalized infidelity that is convex for non-unitary state matrices. Continuity of the state across window boundaries is enforced by equality constraints. A quadratic penalty optimization method is used to solve the constrained optimal control problem, and an efficient adjoint technique is employed to calculate the gradients in each iteration. We demonstrate the effectiveness of the proposed method through numerical experiments on quantum Fourier transform gates in systems with 2, 3, and 4 qubits, noting a speedup of 80x for evaluating the gradient in the 4-qubit case, highlighting the method's potential for optimizing control pulses in multi-qubit quantum systems.

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arXiv2023

A practical approach to determine minimal quantum gate durations using amplitude-bounded quantum controls

We present an iterative scheme to estimate the minimal duration in which a quantum gate can be realized while satisfying hardware constraints on the control pulse amplitudes. The scheme performs a sequence of unconstrained numerical optimal control cycles that each minimize the gate fidelity for a given gate duration alongside an additional penalty term for the control pulse amplitudes. After each cycle, the gate duration is adjusted based on the inverse of the resulting maximum control pulse amplitudes, by re-scaling the dynamics to a new duration where control pulses satisfy the amplitude constraints. Those scaled controls then serve as an initial guess for the next unconstrained optimal control cycle, using the adjusted gate duration. We provide multiple numerical examples that each demonstrate fast convergence of the scheme towards a gate duration that is close to the quantum speed limit, given the control pulse amplitude bound. The proposed technique is agnostic to the underlying system and control Hamiltonian models, as well as the target unitary gate operation, making the time-scaling iteration an easy to implement and practically useful scheme for reducing the durations of quantum gate operations.

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arXiv2023

Deterministic and Bayesian Characterization of Quantum Computing Devices

Motivated by the noisy and fluctuating behavior of current quantum computing devices, this paper presents a data-driven characterization approach for estimating transition frequencies and decay times in a Lindbladian dynamical model of a superconducting quantum device. The data includes parity events in the transition frequency between the first and second excited states. A simple but effective mathematical model, based upon averaging solutions of two Lindbladian models, is demonstrated to accurately capture the experimental observations. A deterministic point estimate of the device parameters is first performed to minimize the misfit between data and Lindbladian simulations. These estimates are used to make an informed choice of prior distributions for the subsequent Bayesian inference. An additive Gaussian noise model is developed for the likelihood function, which includes two hyper-parameters to capture the noise structure of the data. The outcome of the Bayesian inference are posterior probability distributions of the transition frequencies, which for example can be utilized to design risk neutral optimal control pulses. The applicability of our approach is demonstrated on experimental data from the Quantum Device and Integration Testbed (QuDIT) at Lawrence Livermore National Laboratory, using a tantalum-based superconducting transmon device.

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arXiv2023

Mathematical approaches for characterization, control, calibration and validation of a quantum computing device

Quantum computing has received significant amounts of interest from many different research communities over the last few years. Although there are many introductory texts that focus on the algorithmic parts of quantum computing, there is a dearth of publications that describe the modeling, calibration and operation of current quantum computing devices. One aim of this report is to fill that void by providing a case study that walks through the entire procedure from the characterization and optimal control of a qudit device at Lawrence Livermore National Laboratory (LLNL) to the validation of the results. A goal of the report is to provide an introduction for students and researchers, especially computational mathematicians, who are interested in but new to quantum computing. Both experimental and mathematical aspects of this procedure are discussed. We present a description of the LLNL QuDIT testbed, the mathematical models that are used to describe it, and the numerical methods that are used to to design optimal controls. We also present experimental and computational methods that can be used to characterize a quantum device. Finally, an experimental validation of an optimized control pulse is presented, which relies on the accuracy of the characterization and the optimal control methodologies.

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arXiv2022

Noise-specific beats in the higher-level Ramsey curves of a transmon qubit

In the higher levels of superconducting transmon devices, and more generally charge sensitive devices, $T_2^*$ measurements made in the presence of low-frequency time-correlated $1/f$ charge noise and quasiparticle-induced parity flips can give an underestimation of the total dephasing time. The charge variations manifest as beating patterns observed in the overlay of several Ramsey fringe curves, and are reproduced with a phenomenological Ramsey curve model which accounts for the charge variations. $T_2^*$ dephasing times which more accurately represent the total dephasing time are obtained. The phenomenological model is compared with a Lindblad master equation model. Both models are found to be in agreement with one another and the experimental data. Finally, the phenomenological formulation enables a simple method in which the power spectral density (PSD) for the low-frequency noise can be inferred from the overlay of several Ramsey curves.

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arXiv2022

Binary Optimal Control Of Single-Flux-Quantum Pulse Sequences

We introduce a binary, relaxed gradient, trust-region method for optimizing pulse sequences for single flux quanta (SFQ) control of a quantum computer. The pulse sequences are optimized with the goal of realizing unitary gate transformations. Each pulse has a fixed amplitude and duration. We model this process as an binary optimal control problem, constrained by Schrödinger's equation, where the binary variables indicate whether each pulse is on or off. We introduce a first-order trust-region method, which takes advantage of a relaxed gradient to determine an optimal pulse sequence that minimizes the gate infidelity, while also suppressing leakage to higher energy levels. The proposed algorithm has a computational complexity of ${\cal O}(p\log(p)$, where $p$ is the number of pulses in the sequence. We present numerical results for the H and X gates, where the optimized pulse sequences give gate fidelity's better than $99.9\%$, in $\approx 25$ trust-region iterations.

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arXiv2022

Optimal Control of Closed Quantum Systems via B-Splines with Carrier Waves

We consider the optimal control problem of determining electromagnetic pulses for implementing logical gates in a closed quantum system, where the Hamiltonian models the dynamics of coupled superconducting qudits. The quantum state is governed by Schrödinger's equation, which we formulate in terms of the real and imaginary parts of the state vector and solve by the Störmer-Verlet scheme, which is a symplectic partitioned Runge-Kutta method. A novel parameterization of the control functions based on B-splines with carrier waves is introduced. The carrier waves are used to trigger the resonant frequencies in the system Hamiltonian, and the B-spline functions specify their amplitude and phase. This approach allows the number of control parameters to be independent of, and significantly smaller than, the number of time steps for integrating Schrödinger's equation. We present numerical examples of how the proposed technique can be combined with an interior point L-BFGS algorithm for realizing quantum gates, and generalize our approach to calculate risk-neutral controls that are resilient to noise in the Hamiltonian model. The proposed method is also shown to compare favorably with QuTiP/pulse\_optim and Grape-Tensorflow.

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arXiv2021

Quandary: An open-source C++ package for high-performance optimal control of open quantum systems

Quantum optimal control can be used to shape the control pulses for realizing unitary and non-unitary transformations of quantum states. These control pulses provide the fundamental interface between the quantum compiler and the quantum hardware. Most current software for quantum optimal control (e.g. Qutip or Krotov) is restricted to run on shared memory platforms, limiting their applicability to smaller quantum systems, in particular if interactions with the environment are taken into account. This paper gives an overview of the open-source code Quandary, which is designed to solve quantum control problems in larger open quantum systems modelled by Lindblad's master equation. Implemented in C++, Quandary uses the message passing paradigm for distributed memory computers that enables scalability to large numbers of compute cores. Accompanied by numerical examples, this paper presents an overview on existing theoretical developments for open optimal quantum control realizing state-to-state transfer, unitary gate optimization as well as state-preparation, and presents the numerical tools and implementation aspect as realized in Quandary, for deployment on modern high-performance computing platforms.

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arXiv2021

Quantum Optimal Control for Pure-State Preparation Using One Initial State

This paper presents a framework for solving the pure-state preparation problem using numerical optimal control. As an example, we consider the case where a number of qubits are dispersively coupled to a readout cavity. We model open system quantum dynamics using the Markovian Lindblad master equation, driven by external control pulses. The main result of this paper develops a basis of density matrices (a parameterization) where each basis element is a density matrix itself. Utilizing a specific objective function, we show how an ensemble of the basis elements can be used as a single initial state throughout the optimization process - independent of the system dimension. We apply the general framework to the specific application of ground-state reset of one and two qubits coupled to a readout cavity.

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arXiv2021

Elastic wave propagation in curvilinear coordinates with mesh refinement interfaces by a fourth order finite difference method

We develop a fourth order accurate finite difference method for the three dimensional elastic wave equation in isotropic media with the piecewise smooth material property. In our model, the material property can be discontinuous at curved interfaces. The governing equations are discretized in second order form on curvilinear meshes by using a fourth order finite difference operator satisfying a summation-by-parts property. The method is energy stable and high order accurate. The highlight is that mesh sizes can be chosen according to the velocity structure of the material so that computational efficiency is improved. At the mesh refinement interfaces with hanging nodes, physical interface conditions are imposed by using ghost points and interpolation. With a fourth order predictor-corrector time integrator, the fully discrete scheme is energy conserving. Numerical experiments are presented to verify the fourth order convergence rate and the energy conserving property.

math.NA↗

arXiv2021

Quantum Physics without the Physics

This report explains the basic theory and common terminology of quantum physics without assuming any knowledge of physics. It was written by a group of applied mathematicians while they were reading up on the subject. The intended audience consists of applied mathematicians, computer scientists, or anyone else who wants to improve their understanding of quantum physics. We assume that the reader is familiar with fundamental concepts of linear algebra, differential equations, and to some extent the theory of Hilbert spaces. Most of the material can be found in the book by Nielsen and Chuang and in the lecture notes on open quantum systems by Lidar. Another excellent online source of information is Wikipedia, even though most of its articles on quantum physics assume a solid understanding of physics.

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arXiv2020

Discrete Adjoints for Accurate Numerical Optimization with Application to Quantum Control

This paper considers the optimal control problem for realizing logical gates in a closed quantum system. The quantum state is governed by Schrodinger's equation, which we formulate as a time-dependent Hamiltonian system in terms of the real and imaginary parts of the state vector. The system is discretized with the Stormer-Verlet scheme, which is a symplectic partitioned Runge-Kutta method. Our main theoretical contribution is the derivation of a compatible time-discretization of the adjoint state equation, such that the gradient of the discrete objective function can be calculated exactly, at a computational cost of solving two Schrödinger systems, independently of the number of parameters in the control functions. A parameterization of the control functions based on B-splines with built-in carrier waves is also introduced. The carrier waves are used to specify the frequency spectra of the control functions, while the B-spline functions specify their envelope and phase. This approach allows the number of control parameters to be independent of, and significantly smaller than, the number of time steps for integrating Schrodinger's equation. We consider Hamiltonians that model the dynamics of a superconducting multi-level qudit and present numerical examples of how the proposed technique can be combined with the interior point L-BFGS algorithm from the IPOPT package for realizing quantum gates. In a set of test cases, the proposed algorithm is shown to compare favorably with QuTiP/pulse_optim and Grape-Tensorflow.

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arXiv2020

High-fidelity software-defined quantum logic on a superconducting qudit

Nearly all modern solid-state quantum processors approach quantum computation with a set of discrete qubit operations (gates) that can achieve universal quantum control with only a handful of primitive gates. In principle, this approach is highly flexible, allowing full control over the qubits' Hilbert space without necessitating the development of specific control protocols for each application. However, current error rates on quantum hardware place harsh limits on the number of primitive gates that can be concatenated together (with compounding error rates) and remain viable. Here, we report our efforts at implementing a software-defined $0\leftrightarrow2$ SWAP gate that does not rely on a primitive gate set and achieves an average gate fidelity of $99.4\%$. Our work represents an alternative, fully generalizable route towards achieving nontrivial quantum control through the use of optimal control techniques. We describe our procedure for computing optimal control solutions, calibrating the quantum and classical hardware chain, and characterizing the fidelity of the optimal control gate.

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arXiv2020

Energy conservative SBP discretizations of the acoustic wave equation in covariant form on staggered curvilinear grids

We develop a numerical method for solving the acoustic wave equation in covariant form on staggered curvilinear grids in an energy conserving manner. The use of a covariant basis decomposition leads to a rotationally invariant scheme that outperforms a Cartesian basis decomposition on rotated grids. The discretization is based on high order Summation-By-Parts (SBP) operators and preserves both symmetry and positive definiteness of the contravariant metric tensor. To improve accuracy and decrease computational cost, we also derive a modified discretization of the metric tensor that leads to a conditionally stable discretization. Bounds are derived that yield a point-wise condition that can be evaluated to check for stability of the modified discretization. This condition shows that the interpolation operators should be constructed such that their norm is close to one.

math.NA↗

arXiv2019

Fourth order finite difference methods for the wave equation with mesh refinement interfaces

We analyze two types of summation-by-parts finite difference operators for approximating the second derivative with variable coefficient. The first type uses ghost points, while the second type does not use any ghost points. A previously unexplored relation between the two types of summation-by-parts operators is investigated. By combining them we develop a new fourth order accurate finite difference discretization with hanging nodes on the mesh refinement interface. We take the model problem as the two-dimensional acoustic wave equation in second order form in terms of acoustic pressure, and prove energy stability for the proposed method. Compared to previous approaches using ghost points, the proposed method leads to a smaller system of linear equations that needs to be solved for the ghost point values. Another attractive feature of the proposed method is that the explicit time step does not need to be reduced relative to the corresponding periodic problem. Numerical experiments, both for smoothly varying and discontinuous material properties, demonstrate that the proposed method converges to fourth order accuracy. A detailed comparison of the accuracy and the time-step restriction with the simultaneous-approximation-term penalty method is also presented.

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