Trading T gates for dirty qubits in state preparation and unitary synthesis (1812.00954v2)

Abstract: Efficient synthesis of arbitrary quantum states and unitaries from a universal fault-tolerant gate-set e.g. Clifford+T is a key subroutine in quantum computation. As large quantum algorithms feature many qubits that encode coherent quantum information but remain idle for parts of the computation, these should be used if it minimizes overall gate counts, especially that of the expensive T-gates. We present a quantum algorithm for preparing any dimension-$N$ pure quantum state specified by a list of $N$ classical numbers, that realizes a trade-off between space and T-gates. Our scheme uses $\mathcal{O}(\log{(N/\epsilon)})$ clean qubits and a tunable number of $\sim(\lambda\log{(\frac{\log{N}}{\epsilon})})$ dirty qubits, to reduce the T-gate cost to $\mathcal{O}(\frac{N}{\lambda}+\lambda\log{\frac{N}{\epsilon}}\log{\frac{\log{N}}{\epsilon}})$. This trade-off is optimal up to logarithmic factors, proven through an unconditional gate counting lower bound, and is, in the best case, a quadratic improvement in T-count over prior ancillary-free approaches. We prove similar statements for unitary synthesis by reduction to state preparation. Underlying our constructions is a T-efficient circuit implementation of a quantum oracle for arbitrary classical data.

• The paper presents a quantum algorithm that significantly reduces the T-gate count through a trade-off with idle dirty qubits, achieving a quadratic improvement over traditional methods.
• It employs a SelectSwap network to efficiently prepare N-dimensional states and synthesize unitaries, optimizing both circuit depth and resource allocation.
• The approach offers practical and theoretical benefits, enabling more feasible quantum computations in areas such as machine learning, simulation, and quantum chemistry.

Efficient Trade-Offs in Quantum State Preparation: A Professional Overview

The paper "Trading T gates for dirty qubits in state preparation and unitary synthesis" by Guang Hao Low, Vadym Kliuchnikov, and Luke Schaeffer addresses a crucial aspect of quantum computation—efficient synthesis of quantum states and unitaries. The focus is on optimizing the use of $T$-gates, a critical resource in quantum circuits, by leveraging ancillary qubits, specifically dirty qubits, which are partly occupied during computation.

Summary of Contributions

Problem Context

Many quantum algorithms, particularly in fields like machine learning, physics simulation, and linear equation solving, require the synthesis of arbitrary quantum states or unitaries. Typically, this synthesis involves high-cost operations, particularly in terms of $T$-gates, which are non-Clifford gates essential for fault-tolerant quantum computation but are considerably more costly than Clifford gates.

Methodology and Results

The authors propose a quantum algorithm capable of preparing any dimension-$N$ pure quantum state specified by classical data, offering an innovative trade-off between space (qubit usage) and $T$-gates. This methodology employs a quantum circuit construct termed $SelectSwap$ network. Significant results include:

• Reduction in $T$-gate count to $$\mathcal{O}(\frac{N}{\lambda} + \lambda \log{\frac{N}{\epsilon}})$$, marking a quadratic improvement over previous methods without ancillary qubits.
• Flexible utilization of dirty qubits, consuming $\sim (\lambda \log(\frac{\log{N}}{\epsilon}))$ dirty qubits, where $\lambda$ is adjustable based on available qubits and desired improvements.
• Demonstrated optimality of this trade-off through unconditional lower bounds on gate counts, asserting that their algorithm surpasses traditional ancillary-free approaches, particularly with the use of dirty qubits.
• Efficient implementation achieves a $\sqrt{N}$ scaling in $T$-count with ample availability of dirty qubits, minimizing $T$ depth and optimizing circuit performance.
• Extension of this approach to unitary synthesis with applications in quantum chemistry and matrix-complexity techniques.

Implications

Practical Implications

Quantum computing architectures can leverage idle qubits, which encode quantum information but remain inactive in certain computational phases to reduce circuit costs significantly. This capability can make quantum algorithms more feasible and potentially shorten the timeline towards viable quantum computation applications in industry and research.

Theoretical Implications

This paper provides a framework for future investigation into qubit resource utilization, notably how dirty qubits can systematically lower $T$-gate counts. It also prompts deeper exploration into other circuit elements where such trade-offs may exist.

Future Directions

As quantum computation technology advances, attention may shift toward optimizing not only $T$-gate counts but also comprehensive quantum circuit efficiency, including space-time volume considerations and error-resilient state preparations as referenced in ongoing research. This methodology's integration into block-encoding frameworks and QRAM architectures represents an ongoing research trajectory.

In summation, this paper significantly contributes to reducing quantum operation costs, especially for complex quantum algorithms reliant on extensive classical data processing. By optimizing usage of dirty qubits alongside $T$-gate counts and circuit depth, the algorithm offers a pathway to more efficient quantum computing, both theoretically robust and practically impactful.