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Digital-analog quantum simulation of generalized Dicke models with superconducting circuits
Lucas Lamata
TL;DR
Full quantum simulation of generalized Dicke dynamics across light–matter coupling regimes remains desirable. The paper proposes digital-analog protocols for superconducting circuits, and estimates their errors and resource requirements. The authors conclude that proof-of-principle few-qubit experiments may be performed with current technology, while larger simulations depend on improved fidelities and coherence times.
Problem
A full quantum simulation of Dicke dynamics across coupling regimes and with inhomogeneities, biases, and time-dependent couplings is desirable beyond mean-field treatments.
Source: Introduction · p. 1Method
The paper proposes digital-analog simulations of generalized Dicke models with superconducting qubits coupled to a bosonic mode, covering Fermi-Bose, biased, and periodically pulsed variants.
Source: Introduction · p. 1, The Dicke model · p. 3, Biased Dicke model · p. 5Results
The authors estimate digital errors, step and gate requirements, and conclude that proof-of-principle few-qubit experiments may be performed with current technology.
Source: Discussion · pp. 11–12Takeaways & Limitations
Improved gate fidelity and coherence times may allow simulations involving tens of qubits, while scaling to several tens or hundreds may require error correction.
Source: Discussion · p. 12The digital-error norm for the unbounded bosonic operators is evaluated after restricting them to a finite-dimensional domain Hilbert space containing the states in the considered evolution.
Source: Estimation of the digital error · p. 8
Abstract
from arXiv · showhide
We propose a digital-analog quantum simulation of generalized Dicke models with superconducting circuits, including Fermi-Bose condensates, biased and pulsed Dicke models, for all regimes of light-matter coupling. We encode these classes of problems in a set of superconducting qubits coupled with a bosonic mode implemented by a transmission line resonator. Via digital-analog techniques, an efficient quantum simulation can be performed in state-of-the-art circuit quantum electrodynamics platforms, by suitable decomposition into analog qubit-bosonic blocks and collective single-qubit pulses through digital steps. Moreover, just a single global analog block would be needed during the whole protocol in most of the cases, superimposed with fast periodic pulses to rotate and detune the qubits. Therefore, a large number of digital steps may be attained with this approach, providing a reduced digital error. Additionally, the number of gates per digital step does not grow with the number of qubits, rendering the simulation efficient. This strategy paves the way for the scalable digital-analog quantum simulation of many-body dynamics involving bosonic modes and spin degrees of freedom with superconducting circuits.
Introduction
The Dicke model offers a setting for studying light–matter coupling, but simulation across coupling regimes and with inhomogeneities, biases, and time-dependent interactions remains desirable. This paper proposes a superconducting-circuit digital-analog approach to generalized Dicke models, including a single-step circuit built from collective Tavis-Cummings dynamics and collective rotations.
Introduction
- A full quantum simulation across light–matter coupling regimes and with inhomogeneities, biases, and time-dependent couplings is identified as desirable beyond mean-field dynamics.The introduction notes proof-of-principle Dicke superradiance with two superconducting qubits in the weak-coupling regime and a Dicke phase-transition experiment with a superfluid gas.Source: Introduction · p. 1
- The proposal extends superconducting-circuit digital-analog simulation of the Dicke model to Fermi-Bose condensates, biased Dicke models, and periodically pulsed Dicke models.The work also estimates digital error analytically and numerically.Source: Introduction · p. 1
- A single Trotter step uses iterated collective Tavis-Cummings dynamics at alternative qubit detunings, interspersed with collective x-axis rotations.The circuit depicts interacting evolutions and single-qubit gates as red and orange boxes, respectively.Source: Figure 1 · p. 2
- The approach targets ultrastrong and deep-strong coupling regimes that standard atomic systems cannot reach, and its digital-analog implementation may avoid the A2-term issue of direct implementations.The introduction says the A2 effect may in principle be safely neglected in a digital-analog simulation.Source: Introduction · pp. 1–2
Results
The section develops and evaluates digital-analog simulation protocols for Dicke and generalized Dicke models, analyzing digital error and feasibility through numerical simulations with realistic decoherence. It also discusses extensions to phase probing and broader many-body exploration.
The Dicke model
- Dicke-model simulations estimate digital errors, Trotter steps, and gate counts against state-of-the-art superconducting-circuit experiments, and the protocol’s step gate count does not grow with N.The digital-analog decomposition combines collective Tavis-Cummings and anti-Tavis-Cummings dynamics with collective qubit rotations; its leading error is analyzed in detail.Source: The Dicke model · p. 3
Fermi-Bose condensate
- The Fermi-Bose condensate maps exactly to a generalized Dicke model in the unoccupied/doubly occupied subspace, and its dynamics can be simulated exactly across coupling regimes and a wide range of bosonic excitations.The condensate dynamics can be implemented with a fully analog Tavis-Cummings simulator with inhomogeneous qubit detunings and no Trotterization; the finite-bandwidth Tavis-Cummings limit applies when λ ≪ ω,ω_i^0.Source: Fermi-Bose condensate · pp. 3–5
Biased Dicke model
- The biased Dicke protocol adds gates for the qubit bias term, enabling simulation of its effects on behavior including the superradiance phase transition.One implementation uses tunable-coupling qubits to cancel the interaction and collective microwave-driven Y rotations to generate the σ_i^x terms.Source: Biased Dicke model · p. 5
Pulsed Dicke model
- For the periodically pulsed Dicke model, square-pulse approximations recover delta pulses in the α → ∞, τ → 0, ατ = 1 limit without increasing digital error.Fast coupling switching and single-qubit rotations permit many digital steps; each step uses two Dicke-model evolutions with different couplings, each built from Tavis-Cummings and anti-Tavis-Cummings interactions.Source: Pulsed Dicke model · pp. 5–6
Estimation of the digital error
- With bounded bosonic excitations, the leading digital-error contribution scales at most quadratically with qubit number N, while the Dicke protocol uses a number of gates per step independent of N.The numerical simulations indicate feasibility in analyzed cases where bosonic excitations do not grow much; the Fermi-Bose condensate is expected to have similar error scaling, the bias adds a subdominant term linear in N, and the pulsed model has εPulse(Hi) ≃ 2ε(H1,H2).Source: Estimation of the digital error · pp. 8–9
Numerical simulations
- Numerical simulations compare ideal exact evolution with digitized Dicke and pulsed-Dicke evolution under realistic decoherence modeled by cavity damping, spontaneous emission, and dephasing.The digitized dynamics are applied successively for each Trotter-step component, including Tavis-Cummings evolutions and local gates.Source: Numerical simulations · pp. 9–10
- With 11 Trotter steps (versus about 90 experimentally achieved), N=2 and N=3 fidelities remain large relative to the random-state classical limit during sizable deep-strong-coupling evolution, though fidelity falls with N.The initial free-Hamiltonian ground state changes significantly in this regime, which includes counterrotating terms.Source: Numerical simulations · p. 10
- In the ultrastrong-coupling regime, N=3 produces a larger photon emission rate than N=2, consistent with the larger qubit number.The comparison uses the free-Hamiltonian ground state as the initial state.Source: Numerical simulations · p. 10
- With n=13 Trotter steps, pulsed-model fidelity remains large relative to the classical limit during sizable deep-strong-coupling evolution, decreasing from N=2 to N=3.The pulsed model is the most complex proposed model in total gates and entangling gates per Trotter step.Source: Numerical simulations · pp. 10–11, Figure 7 · p. 11
- Because the pulsed Dicke model is the most gate-intensive proposed model, the Fermi-Bose and biased models are expected to need fewer Trotter steps for similar fidelity.Source: Numerical simulations · p. 11
- The simulations are presented as feasible with current circuit-QED technology, with digitized adiabatic evolution, photon-count benchmarking, and DAQS exploration of exotic phases proposed as extensions.The passage notes that low- or no-symmetry cases may pose a sign problem for classical computers, and suggests a simulator with a few tens of qubits may solve such dynamics exponentially faster; the protocols may also generate highly entangled multipartite states.Source: Numerical simulations · p. 11
Discussion
The discussion summarizes the proposed generalized Dicke-model simulation and its estimates of digital errors, steps, and gates. It states that few-qubit proof-of-principle experiments are feasible with current technology, while larger-scale implementations remain prospective.
Discussion
- The work estimates digital errors, required digital steps, and gates, and compares them with state-of-the-art superconducting-circuit experiments.These estimates include comparisons with experiments having optimal coherence times.Source: Discussion · pp. 11–12
- Proof-of-principle few-qubit experiments across a wide range of coupling regions may be performed with current technology.Source: Discussion · pp. 11–12
- Scaling to several tens or hundreds of qubits may require error correction, while related digital-analog simulations could also be explored in Penning traps.The paper presents these as possibilities for future work, not demonstrated results.Source: Discussion · p. 12