Tal Schwartzman of ITAMP, Center for Astrophysics | Harvard & Smithsonian, and colleagues have devised new unitary protocols for preparing ground states in quantum systems by leveraging multiple copies of the system alongside controlled-SWAP operations. Their work addresses a key challenge in quantum computation: efficiently achieving low-energy state preparation, a goal crucial for both simulation and computation.

The researchers detail two circuit designs, one offering provable polynomial-in-depth convergence but rapidly growing width, and a more scalable “hedge” architecture, and demonstrate that mid-circuit post-selection can accelerate the process with achievable probabilities. This approach outlines how hybrid analog-digital circuits can complement existing state-preparation methods in the near term.

Imaginary Time Evolution for Ground State Preparation

The team constructed circuits approximating imaginary time evolution, a technique that suppresses higher-energy states, and demonstrated polynomial-in-depth convergence. This improvement signifies a potential pathway toward more efficient quantum computation and simulation by refining how initial quantum states are established. The circuits developed rely on real-time evolution applied to each copy of the system alongside controlled-SWAP operations that mediate interactions between them; this combination allows for deterministic, unitary protocols that approximate the effects of imaginary time evolution.

Researchers observed that the performance of these circuits, specifically in ground state preparation, can be further enhanced through mid-circuit post-selection, a process where unsuccessful attempts are discarded, indicating a viable strategy for improving efficiency despite the inherent probabilistic nature of quantum processes. Numerical simulations, using matrix product states, confirm this convergence as the number of copies increases.

Two distinct circuit architectures were explored, each presenting a different trade-off between resource demands and convergence speed. One offers provable polynomial-in-depth convergence, meaning the circuit’s accuracy improves predictably with its depth, but requires a rapidly increasing number of qubits as the system scales. Conversely, a “hedge” architecture achieves comparable accuracy with only a polynomial increase in qubit count, suggesting a more scalable approach, although this is currently supported by numerical evidence rather than formal proof.

The researchers report that “for imaginary time evolution, as can be seen in the log-log plot of d, the infidelity with the exact state scales as a power law in regions where n is large enough,” detailing the observed scaling behavior. These hybrid analog-digital circuits, leveraging multi-copy registers and SWAP-mediated couplings, represent a complementary approach to existing state-preparation methods, potentially enabling near-term advancements in quantum technologies.

Unitary Multi-Copy Protocols Approximate Imaginary Time

Hannes Pichler of Quantum Optics and Quantum Information at the Austrian Academy of Sciences, alongside colleagues, are refining methods for preparing quantum states by exploring the trade-offs between circuit speed and qubit requirements. Their work centers on two distinct circuit designs, a “tree” and a “hedge” architecture, each approaching the challenge of ground state preparation with a different strategy for managing computational resources. The tree architecture guarantees convergence, meaning its accuracy predictably improves with increased computational depth, but demands an exponentially growing number of qubits as that depth increases.

The tree circuit achieves this with provable polynomial-in-depth convergence; the authors state the error decreases with increasing depth. However, the number of qubit copies required scales exponentially with depth, making it increasingly resource-intensive for larger systems. Numerical analysis reveals the worst-case upper bound scales exponentially with both a parameter β and the system Hamiltonian |H|, further emphasizing the resource demands.

Researchers found that, for physically relevant systems, the scaling with system size can be more favorable than the worst-case scenario, offering some potential for optimization. Simulations suggest this design, while heuristic, meaning its convergence isn’t mathematically guaranteed, achieves performance similar to the tree architecture without the same exponential scaling of qubit resources. This offers a potentially more scalable path toward building practical quantum computers.

Hedge Circuit Achieves Polynomial Width Accuracy

The challenge of scaling quantum computers hinges on efficiently preparing the low-energy states needed for complex calculations, and new circuit designs are addressing this critical hurdle. Researchers are now distinguishing between approaches that prioritize provable accuracy and those that offer potentially greater scalability. The analysis focuses on two concrete circuit families: a tree architecture with provable polynomial-in-depth convergence but rapidly growing width, and a compact “hedge” architecture that achieves comparable accuracy with only polynomial width in a heuristic construction supported by numerics.

Mid-Circuit Post-Selection Accelerates Convergence

The conventional expectation that achieving accurate ground state preparation in quantum systems demands substantial coherent depth and resources is being challenged by new protocols leveraging multiple qubit copies and a technique called mid-circuit post-selection. Analysis of these circuits reveals that discarding unsuccessful computational attempts, post-selection, can significantly accelerate convergence toward the desired low-energy state, even with reasonable probabilities of success.

While the worst-case error bound for the tree architecture scales exponentially with β and the system Hamiltonian |H|, numerical results suggest that for systems mirroring physical realities, the scaling can be considerably more favorable. The incorporation of post-selection provides a strategy for optimization, suggesting that even with inherent probabilistic elements, quantum computations can be optimized for efficiency through strategic mid-circuit evaluation and filtering of results, opening avenues for near-term implementation using hybrid analog-digital circuits and readily available multi-copy registers.

Circuit Volume Trade-off for Ground State Estimation

Achieving faster convergence in quantum ground state preparation is now possible by strategically sacrificing circuit complexity, according to new findings. This technique allows for a trade-off between the total number of gates used in a circuit and the number of measurements needed to estimate ground state observables, effectively shifting computational resources. Numerical evidence indicates that infidelity with the ideal imaginary evolved state decreases as the number of copies increases.

The ability to interchange circuit volume with the number of measurements represents a significant optimization for resource-constrained quantum systems. The protocols require multiple copies of the system, real-time evolution under the system Hamiltonian, and controlled-SWAP operations, which are naturally suited to platforms with multi-copy registers and SWAP-mediated couplings. The researchers note that “these hybrid analog-digital protocols can be implemented with existing quantum simulation platforms,” both to refine existing methods and to explore thermal behavior.

SWAP-Mediated Couplings Enable Platform Implementation

Mid-circuit post-selection offers a pathway to faster convergence in ground state preparation protocols, with simulations demonstrating achievable success probabilities for these techniques. This acceleration stems from the ability to discard unsuccessful attempts during computation, effectively refining the process without requiring substantial increases in computational resources. Superconducting qubit quantum computers and trapped-ion devices already demonstrate the universal control needed to execute these SWAP operations, synthesizing them digitally on isolated sub-systems functioning as independent copies.

These platforms also natively enable the required analog Hamiltonian evolution, as evidenced by recent hybrid digital-analog experiments. The practicality of this approach is further enhanced by the natural availability of multi-copy registers and SWAP-mediated couplings in these leading quantum architectures. This flexibility allows for precision to be added where needed, potentially opening avenues for more efficient and scalable quantum simulations. The demonstrated trade-offs represent a step toward realizing practical quantum computation with near-term technologies.

Ground State Preparation Complements Existing Methods

Researchers are finding ways to refine existing quantum state preparation techniques through hybrid analog-digital circuits, offering a path toward more practical quantum computation with near-term technology. This approach doesn’t require entirely new hardware, but instead builds upon the capabilities of current quantum simulation platforms. This means designers can reduce the complexity of the quantum circuit itself by accepting a greater need for computational resources after the circuit runs, and vice versa.

The protocols can be integrated with approaches like adiabatic state preparation, further increasing ground state fidelities, and can also be used to study thermal behavior. The research highlights the potential for leveraging the strengths of both analog and digital quantum systems.

👉 More information
🗞 Imaginary Time Evolution and Ground State Preparation Using Unitary Multi-Copy Protocols
✍️ Tal Schwartzman, Torsten V. Zache, Hannes Pichler and H. R. Sadeghpour
🧠 DOI: http://link.aps.org/doi/10.1103/1tf6-bc55

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