Diamond-shaped quantum circuit for real-time quantum dynamics in one dimension

At a Glance

Metadata	Details
Publication Date	2024-12-26
Journal	Physical Review Research
Authors	Shohei Miyakoshi, Takanori Sugimoto, Tomonori Shirakawa, Seiji Yunoki, Hiroshi Ueda
Institutions	RIKEN Center for Emergent Matter Science, RIKEN Center for Computational Science
Citations	2
Analysis	Full AI Review Included

Executive Summary

The research introduces and validates a highly efficient quantum circuit architecture—the diamond-shaped quantum circuit—designed for simulating long-time, highly entangled quantum dynamics, particularly relevant for noisy intermediate-scale quantum (NISQ) and future fault-tolerant quantum computers (FTQCs).

High Fidelity for Long-Time Dynamics: The diamond circuit accurately represents the numerically exact time-evolved state for the transverse-field Ising model (h = 0) over long periods, where classical simulation is challenging.
Superior Efficiency: For the integrable case (h = 0), the diamond circuit achieved an infidelity approximately 500 times smaller than a sequential-type circuit with a comparable number of two-qubit gates.
Parameter Compression: The diamond circuit significantly reduces the internal degrees of freedom compared to the single-layer multiqubit circuit required for exact representation. For a system size L=11, the parameter compression ratio is approximately 35.6.
Volume Law Compliance: Structurally, the diamond-shaped ansatz satisfies the entropic volume law, which is a necessary condition for representing the highly entangled states generated during non-equilibrium dynamics.
Scalability Advantage: By decomposing large multiqubit gates into a sparse network of nearest-neighbor two-qubit gates, the circuit is easier to implement on current hardware, minimizing gate count and reducing noise accumulation.
Limitation Identified: The performance advantage of the diamond circuit diminishes rapidly when simulating chaotic dynamics induced by a large longitudinal field (h > 0.1), suggesting limitations in representing complex quantum information scrambling.

Technical Specifications

The following specifications relate to the simulation parameters and performance metrics of the quantum circuit ansatze.

Parameter	Value	Unit	Context
Model System	1D Quantum Ising Chain	N/A	Hamiltonian includes transverse (g) and longitudinal (h) fields.
Trotter Decomposition Order	Second-order	N/A	Used for time evolution operator V(Δt).
Time Step (JΔt)	0.01	J^-1	Chosen to ensure negligible Trotter error in numerical results.
System Size (L) Tested	11 and 16	Qubits	Benchmarking expressivity and scalability.
Multiqubit Gate Size (l) for Exact State	[L/2] + 1	Qubits	Required for single-layer sequential circuit to represent any arbitrary state.
Diamond Circuit Gate Count (L=11)	30	Two-qubit gates (U(4))	Calculated using n_g = (L² - 1) / 4 (for L odd).
Parameter Compression Ratio (L=11)	~35.6	N/A	Reduction in independent real parameters (15,487 / 435).
Infidelity Target (ε_a)	10^-14	N/A	Absolute convergence threshold for optimization.
Relative Convergence Threshold (ε_r)	10^-4	N/A	Relative convergence threshold for optimization.
Fidelity Improvement (h=0, Jt > 3)	~500 times smaller infidelity	N/A	Diamond circuit vs. three-layer sequential circuit (L=11).
Longitudinal Field (h) Performance Limit	> 0.1	N/A	Diamond circuit advantage is lost in this nonintegrable regime.

Key Methodologies

The real-time quantum dynamics were simulated using a variational approach based on the Quantum Circuit Encoding (QCE) algorithm, adapted for time evolution.

Hamiltonian and Time Evolution: The 1D quantum Ising model Hamiltonian (Ĥ) was used. Time evolution was implemented via a second-order Trotter decomposition of the unitary operator e^-iĤΔt, resulting in the time-step operator V(Δt).
Variational Ansatz Selection: Three primary circuit types were tested as variational ansatze (Ĉ_t):
- Single-layer sequential-type (composed of large l-qubit gates).
- M-layer sequential-type (composed of two-qubit gates).
- Diamond-shaped circuit (a sparse, two-qubit gate approximation of the multiqubit circuit).
Optimization Objective: The circuit parameters were iteratively optimized to maximize the fidelity (overlap) I(t) between the approximate state |Φ(t)〉 = Ĉ_t|Φ〉 and the numerically exact time-evolved state |Ψ(t)〉 = V(Δt)|Ψ(t - Δt)〉.
Local Optimization (QCE Adaptation): Optimization was performed sequentially, gate by gate, using a local update rule derived from the Singular Value Decomposition (SVD) of the relevant tensor (similar to tensor network methods).
Sweep Iteration: The local optimization process was repeated through multiple “sweep updates” (left-to-right, then right-to-left) until the fidelity converged based on predefined absolute (ε_a) and relative (ε_r) thresholds, within a minimum (w_min) and maximum (w_max) number of sweeps.
Initial Condition: The optimized circuit from the previous time step (Ĉ_t-Δt) was used as the initial condition for optimizing the current time step (Ĉ_t), ensuring continuous tracking of the dynamics.

Commercial Applications

The development of highly efficient, low-depth quantum circuits like the diamond ansatz has direct implications for optimizing quantum computation resources.

Quantum Simulation of Materials: Enables the efficient simulation of non-equilibrium dynamics in condensed matter systems (like the Ising model) that exhibit complex, volume-law entanglement, which is intractable for classical tensor network methods.
NISQ Device Optimization: The diamond circuit’s structure, which uses a minimal number of nearest-neighbor two-qubit gates, is highly suitable for implementation on current noisy quantum hardware, reducing the impact of decoherence and noise.
Quantum Algorithm Design: Provides a blueprint for constructing compact, high-expressivity ansatze for variational quantum algorithms (e.g., VQE, QAOA) where minimizing circuit depth and gate count is critical for achieving quantum advantage.
Error Mitigation Techniques: Compact circuits with minimal gate operations enhance the effectiveness of error mitigation techniques (such as Zero-Noise Extrapolation or Probabilistic Error Cancellation) by reducing the overall noise accumulation.
Classification of Quantum Dynamics: The methodology offers a path toward classifying different quantum dynamical systems based on the optimal quantum circuit structure required to describe them (e.g., diamond-shaped for integrable systems vs. deeper circuits for chaotic systems).

View Original Abstract

In recent years, quantum computing has evolved as an exciting frontier, with the development of numerous algorithms dedicated to constructing quantum circuits that adeptly represent quantum many-body states. However, this domain remains in its early stages and requires further refinement to better understand the effective construction of highly entangled quantum states within quantum circuits. Here, we demonstrate that quantum many-body states can be universally represented using a quantum circuit comprising multiqubit gates. Furthermore, we evaluate the efficiency of a quantum circuit constructed with two-qubit gates in quench dynamics for the transverse-field Ising model. In this specific model, despite the initial state being classical without entanglement, it undergoes long-time evolution, eventually leading to a highly entangled quantum state. Our results reveal that a diamond-shaped quantum circuit, designed to approximate the multiqubit gate-based quantum circuit, remarkably excels in accurately representing the long-time dynamics of the system. Moreover, the diamond-shaped circuit follows the volume law behavior in entanglement entropy, offering a significant advantage over alternative quantum circuit constructions employing two-qubit gates. Published by the American Physical Society 2024

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Original Source

DOI: https://doi.org/10.1103/physrevresearch.6.043318