Efficient synthesis of probabilistic quantum circuits with fallback (1409.3552v2)

Abstract: Recently it has been shown that Repeat-Until-Success (RUS) circuits can approximate a given single-qubit unitary with an expected number of $T$ gates of about $1/3$ of what is required by optimal, deterministic, ancilla-free decompositions over the Clifford+$T$ gate set. In this work, we introduce a more general and conceptually simpler circuit decomposition method that allows for synthesis into protocols that probabilistically implement quantum circuits over several universal gate sets including, but not restricted to, the Clifford+$T$ gate set. The protocol, which we call Probabilistic Quantum Circuits with Fallback (PQF), implements a walk on a discrete Markov chain in which the target unitary is an absorbing state and in which transitions are induced by multi-qubit unitaries followed by measurements. In contrast to RUS protocols, the presented PQF protocols terminate after a finite number of steps. Specifically, we apply our method to the Clifford+$T$, Clifford+$V$, and Clifford+$\pi/12$ gate sets to achieve decompositions with expected gate counts of $\log_b(1/\varepsilon)+O(\log(\log(1/\varepsilon)))$, where $b$ is a quantity related to the expansion property of the underlying universal gate set.

• The paper introduces the PQF protocol which provides a deterministic fallback mechanism to complete quantum circuit synthesis in a fixed number of steps.
• The approach extends synthesis techniques to various gate sets, including Clifford+T, Clifford+V, and Clifford+7/12, ensuring broad applicability and precision.
• The PQF method reduces expected gate counts and improves algorithmic efficiency, offering a scalable alternative to traditional deterministic and RUS strategies.

Analysis of Efficient Probabilistic Quantum Circuits with Fallback

The paper "Efficient Synthesis of Probabilistic Quantum Circuits with Fallback" by Alex Bocharov, Martin Roetteler, and Krysta M. Svore presents an innovative approach for the decomposition of quantum circuits into probabilistic processes. The method improves on existing techniques, specifically the Repeat-Until-Success (RUS) circuits, by providing a conceptually simpler and more generalizable decomposition method applicable to several universal gate sets such as Clifford+T, Clifford+V, and Clifford+7/12. The proposed approach is encapsulated in the Probabilistic Quantum Circuits with Fallback (PQF) framework.

Core Contributions

1. Probabilistic Circuit Design: The PQF protocol introduces a deterministic fallback mechanism. In contrast to RUS circuits, which may require an unbounded number of trials, PQF circuits are designed to terminate after a fixed finite number of steps unless further iterations are beneficial in terms of cost-effectiveness. This design is built on a discrete Markov chain foundation where the termination is guaranteed by the inherent structure.
2. Gate Set Flexibility: While the RUS method is constrained primarily to the Clifford+T basis, PQF extends to various gate sets, enhancing its applicability in different quantum computational contexts. Notably, it applies to the Clifford+V and Clifford+7/12 bases. Each base has its peculiarities and PQF adapts to these while ensuring high precision.
3. Improved Cost Efficiency: Through the probabilistic synthesis procedure, PQF achieves a lower expected gate count compared to deterministic strategies. This reduction is achieved without incurring the high complexity of traditional techniques like the Solovay-Kitaev algorithm, thus positioning PQF circuits as both cost-effective and scalable.
4. Algorithmic Efficiency: The detailed algorithmic design allows for an anticipated gate count that surpasses the standards set by existing unitary circuit synthesis, such as the deterministic ancilla-free methods. The expected T-count for PQF circuits maintains a tight bound around log(1/ε), emphasizing the protocol's efficiency in resource utilization.

Numerical Results

The paper includes substantial numerical validation of the PQF synthesis method. Across multiple scenarios and universal gate bases, the experiments demonstrate the robustness and reliability of PQF, yielding consistent reductions in expected gate counts. This empirical evidence not only supports theoretical claims but also provides a practical viewpoint on implementation viability.

Theoretical and Practical Implications

The findings within this paper indicate several implications for both theoretical advancements and practical implementations in quantum computing:

• Quantum Algorithm Compilation: The PQF approach provides a compelling tool for the efficient compilation of quantum algorithms. Its adaptability across various gate sets means it can potentially unify different synthesis approaches.
• Quantum Computing Architectures: By reducing the expected gate counts, PQF circuits could significantly lower the resource requirements for fault-tolerant quantum computation, facilitating more complex quantum algorithms with existing hardware capabilities.
• Extended Application Scope: Given its flexibility, PQF can handle more complex unitary operations, indicating potential for broader applications in multi-qubit quantum systems.

Future Directions

The paper suggests future research directions focused on further generalization to multi-qubit systems and other universal bases. These expansions could reinforce PQF's role as a universal solution in quantum circuit synthesis. Additionally, addressing open questions regarding formal proofs of underlying mathematical conjectures could fortify the theoretical foundation of the PQF framework.

In conclusion, the PQF method represents a substantial step forward in quantum circuit synthesis, providing an efficient, adaptable, and theoretically sound approach for building quantum circuits across multiple gate sets.