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IEEE Transactions on Quantum Engineering

Emulation of Density Matrix Dynamics With Classical Analog Circuits

A. J. Cressman, Rahul Sarpeshkar

Published Jan 1, 2025• 1 citation
Publisher

Abstract

Analog circuits have emerged as a valuable quantum emulation and simulation platform. Specifically, they have been experimentally shown to excel in emulating coherent state vector dynamics and motifs of quantum circuits, such as the quantum Fourier transform, tensor product superpositions, two-level systems such as Josephson junctions, and nuclear magnetic resonance state dynamics, all on a very large scale integration chip at room temperature (Cressman et al., 2022; Sarpeshkar, 2019a, 2019b, 2019c; Sarpeshkar, 2020). However, the ability to model simple state vectors is insufficient for modeling open quantum systems, i.e., systems with environmental noise. Noisy quantum systems are essential in practical implementations and applications that exploit noise. The density matrix formalism enables us to model such states, including finite reservoir state systems, and all states that can be represented as state vectors. To our knowledge, no one has yet demonstrated the mapping of a density matrix system to classical analog circuit components. We review the procedure for emulating the dynamics of a finite state vector with four essential analog circuit components and extend this procedure to emulate density matrix dynamics. We then simulate these systems as analog circuits in the presence of noise. This protocol opens up exciting possibilities for further research and development in noisy quantum emulation and simulation using analog circuits for arbitrarily large or small systems.

DOI
10.1109/tqe.2025.3552736

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Analog circuits have emerged as a valuable quantum emulation and simulation platform. Specifically, they have been experimentally shown to excel in emulating coherent state vector dynamics and motifs of quantum circuits, such as the quantum Fourier transform, tensor product superpositions, two-level systems such as Josephson junctions, and nuclear magnetic resonance state dynamics, all on a very large scale integration chip at room temperature (Cressman et al., 2022; Sarpeshkar, 2019a, 2019b, 2019c; Sarpeshkar, 2020). However, the ability to model simple state vectors is insufficient for modeling open quantum systems, i.e., systems with environmental noise. Noisy quantum systems are essential in practical implementations and applications that exploit noise. The density matrix formalism enables us to model such states, including finite reservoir state systems, and all states that can be represented as state vectors. To our knowledge, no one has yet demonstrated the mapping of a density matrix system to classical analog circuit components. We review the procedure for emulating the dynamics of a finite state vector with four essential analog circuit components and extend this procedure to emulate density matrix dynamics. We then simulate these systems as analog circuits in the presence of noise. This protocol opens up exciting possibilities for further research and development in noisy quantum emulation and simulation using analog circuits for arbitrarily large or small systems.
Quantum dynamical systems are capable of powerful computation but are hard to emulate on digital computers. We show that four novel analog circuit parts can emulate the phase-coherent unitary dynamics of such systems. These four parts are: a Planck capacitance analogous to a neuronal membrane capacitance; a quantum admittance element, together with the Planck capacitance, analogous to a neuronal quadrature oscillator; a quantum transadmittance element analogous to a complex neuronal synapse; and a quantum transadmittance mixer element analogous to a complex neuronal synapse with resonant modulation. These parts may be emulated classically, with paired real-value voltages on paired Planck capacitances corresponding to the real and imaginary portions of a probability amplitude; and appropriate paired real-value currents onto these Planck capacitances corresponding to diagonal (admittance), off-diagonal (transadmittance), or controlled off-diagonal (transadmittance mixer) Hamiltonian energy terms. The superposition of 2n simultaneously phase-coherent and symmetric probability-voltage amplitudes with O(n) of these parts, in a tensor-product architecture enables analog emulation of the quantum Fourier transform (QFT). Implementation of our circuits on an analog integrated circuit in a 0.18 μm process yield experimental results consistent with mathematical theory and computer simulations for emulations of NMR, Josephson junction, and QFT dynamics. Our results suggest that linear oscillatory neuronal networks with pairs of complex subthreshold/nonspiking sine and cosine neurons that are coupled together via complex synapses to other such complex neurons can architect quantum-inspired computation with classical analog circuits. Thus, an analog-circuit mapping between quantum and neural computation, both of which exploit analog computation for powerful operation, can enable future synergies between these fields.
Simulating quantum circuits with classical computers requires resources growing exponentially in terms of system size.Real quantum computer with noise, however, may be simulated polynomially with various methods considering different noise models.In this work, we simulate random quantum circuits in one dimension with matrix product density operators (MPDOs), for different noise models such as dephasing, depolarizing, and amplitude damping.We show that the method based on matrix product states (MPSs) fails to approximate the noisy output quantum states for any of the noise models considered, while the MPDO method approximates them well.Compared with the method of matrix product operators (MPOs), the MPDO method reflects a clear physical picture of noise (with inner indices taking care of the noise simulation) and quantum entanglement (with bond indices taking care of two-qubit gate simulation).Consequently, in case of weak system noise, the resource cost of the MPDO will be significantly less than that of the MPO due to a relatively small inner dimension needed for the simulation.In case of strong system noise, a relatively small bond dimension may be sufficient to simulate the noisy circuits, indicating a regime that the noise is large enough for an "easy" classical simulation, which is further supported by a comparison with the experimental results on an IBM cloud device.Moreover, we propose a more effective tensor updates scheme with optimal truncations for both the inner and the bond dimensions, performed after each layer of the circuit, which enjoys a canonical form of the MPDO for improving simulation accuracy.With truncated inner dimension to a maximum value κ and bond dimension to a maximum value χ , the cost of our simulation scales as ∼NDκ 3 χ 3 , for an N-qubit circuit with depth D.
Demonstrations of quantum advantage for certain sampling problems have generated considerable excitement for quantum computing and have further spurred the development of circuit-model quantum computers, which represent quantum programs as a sequence of quantum gates acting on a finite number of qubits. Amongst this excitement, analog quantum computation has become less prominent, with the expectation that circuit-model quantum computers will eventually be sufficient for emulating analog quantum computation and thus rendering analog quantum computation obsolete. In this work we explore the basic requirements for emulating a specific analog quantum computation in the circuit model: the preparation of a biased superposition of degenerate ground states of an Ising Hamiltonian using an adiabatic evolution. We show that the overhead of emulation is substantial even for this simple problem. This supports using analog quantum computation for solving time-dependent Hamiltonian dynamics in the short term and midterm, assuming analog errors can be made low enough and coherence times long enough to solve problems of practical interest.
We discuss a new approach to simulate quantum algorithms using classical probabilistic bits and circuits. Each qubit (a two-level quantum system) is initially mapped to a vector in an eight dimensional probability space (equivalently, to a classical random variable with eight probabilistic outcomes). The key idea in this mapping is to store both the amplitude and phase information of the complex coefficients that describe the qubit state in the probabilities. Due to the identical tensor product structure of combining multiple quantum systems as well as multiple probability spaces, nnn qubits are then mapped to a tensor product of nnn 8-dimensional probabilistic vectors (i.e., the Hilbert space of dimension 2n2^n2n is mapped to a probability space of dimension 8n8^n8n). After this initial mapping, we show how to implement the analogs of single-qubit and two-qubit gates in the probability space using correlation-inducing operations on these classical random variables. The key defining feature of both the mapping to the probability space and the transformations in this space (i.e., operations on the random variables) is that they are not linear, but instead affine. Using this architecture, the evolution of the 2n2^n2n complex coefficients of the quantum system can be tracked in the joint fully-correlated probabilities of the polynomial number of random variables. We then give specific procedures for implementing (1) the Deutsch-Jozsa algorithm, and (2) the Quantum Fourier Transform in the probability space. Identical to the Quantum case, simulating the Quantum Fourier Transform in the probability space requires O(n)O(n)O(n) probabilistic bits and O(n2)O(n^2)O(n2) (i.e., quadratic in the number of quantum bits) operations.
This study presents a benchmark of quantum noise scaling in GHZ circuits using PennyLane-based simulations. The experiments investigate how increasing the number of qubits and applying noise affect GHZ-state fidelity, while also comparing the computational cost of noisy quantum simulation with classical density-matrix simulation. The benchmark evaluates circuit sizes across multiple qubit counts under controlled simulation conditions and records the resulting state fidelity and computational behavior. The results provide an empirical view of the relationship between qubit scaling, noise, and classical simulation cost. In particular, the study highlights the increasing computational burden associated with density-matrix representations as system size grows. The accompanying implementation, experimental data, and visualization scripts are provided as open-source materials to support reproducibility. The results are intended as a practical benchmark for studying the scaling behavior of noisy quantum circuits on classical simulation backends.
Understanding the effects of noise on quantum computations is fundamental to the development of quantum hardware and quantum algorithms. Simulation tools are essential for quantitatively modelling these effects, yet unless artificial restrictions are placed on the circuit or noise model, accurately modelling noisy quantum computations is an extremely challenging task due to unfavourable scaling of required computational resources. Tensor network methods offer a viable solution for simulating computations that generate limited entanglement or that have noise models which yield low gate fidelities. However, in the most interesting regime of entangling circuits (with high gate fidelities) relevant for error correction and mitigation tensor network simulations often achieve poor accuracy. In this work we develop and numerically test a method for significantly improving the accuracy of tensor network simulations of noisy quantum circuits in the low-noise (i.e. high gate-fidelity) regime. Our method comes with the advantages that it (i) allows for the simulation of quantum circuits under generic types of noise model, (ii) is especially tailored to the low-noise regime, and (iii) retains the benefits of tensor network scaling, enabling efficient simulations of large numbers of qubits. We build upon the observations that adding extra noise to a quantum circuit makes it easier to simulate with tensor networks, and that the results can later be reliably extrapolated back to the low-noise regime of interest. These observations form the basis for a novel emulation technique that we call non-zero noise extrapolation, in analogy to the quantum error mitigation technique of zero-noise extrapolation.