Trading T gates for dirty qubits in state preparation and unitary synthesis
1Quantum Architectures and Computation, Microsoft Research, Washington, Redmond, USA
2Azure Quantum, Microsoft, Washington, Redmond, USA
3Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA
4Joint Center for Quantum Information and Computer Science, University of Maryland, Maryland, College Park, USA
Published: | 2024-06-17, volume 8, page 1375 |
Eprint: | arXiv:1812.00954v2 |
Doi: | https://doi.org/10.22331/q-2024-06-17-1375 |
Citation: | Quantum 8, 1375 (2024). |
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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.
![](https://quantum-journal.org/wp-content/uploads/2024/06/cover.png)
Featured image: Our key result is a realization of any lookup table with $N$ addresses to $b$-bit words by our `SelectSwap' quantum circuit that uses as few as $\mathcal{O}(\sqrt{N b})$ non-Clifford T gates.
TQC 2019
Popular summary
Often, interesting computations require a very large number of queries on a very large amount of classical data. Such large computations necessitate fault-tolerant quantum computation on logical qubits, which have a cost model where Clifford gates are cheap but non-Clifford gates are expensive. In contrast, physical qubits typically have a cost model with cheap single-qubit non-Clifford gates but expensive two-qubit Clifford gates.
Motivated by the constraints of fault-tolerant quantum computation, we present a family of quantum circuits that realize arbitrary quantum oracles with the fewest number of non-Clifford gates, up to an optimal square-root factor improvement over prior art. This work enables more accurate costing of quantum algorithms in the fault-tolerant regime.
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The above citations are from SAO/NASA ADS (last updated successfully 2024-09-02 17:10:33). The list may be incomplete as not all publishers provide suitable and complete citation data.
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