A Meet-in-the-Middle Algorithm for Fast Synthesis of Depth-Optimal Quantum Circuits

Amy, Matthew; Maslov, Dmitri; Mosca, Michele; Roetteler, Martin

doi:10.1109/tcad.2013.2244643

Cited by 433 publications

(437 citation statements)

References 37 publications

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“…One can easily check that the multiple control CNOT gate Λ kþ1 ðXÞ can be implemented using OðkÞ Toffoli gates. Furthermore, the Toffoli gate can be implemented using seven T gates [32,33]. Thus, θ k has T count OðkÞ and, therefore, ϕ n has T count Oð P n k¼1 kÞ ¼ Oðn 2 Þ.…”

Section: Appendix B: Stabilizer Decompositionsmentioning

confidence: 99%

Trading Classical and Quantum Computational Resources

2016

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We propose examples of a hybrid quantum-classical simulation where a classical computer assisted by a small quantum processor can efficiently simulate a larger quantum system. First, we consider sparse quantum circuits such that each qubit participates in Oð1Þ two-qubit gates. It is shown that any sparse circuit on n þ k qubits can be simulated by sparse circuits on n qubits and a classical processing that takes time 2 OðkÞ polyðnÞ. Second, we study Pauli-based computation (PBC), where allowed operations are nondestructive eigenvalue measurements of n-qubit Pauli operators. The computation begins by initializing each qubit in the so-called magic state. This model is known to be equivalent to the universal quantum computer. We show that any PBC on n þ k qubits can be simulated by PBCs on n qubits and a classical processing that takes time 2OðkÞ polyðnÞ. Finally, we propose a purely classical algorithm that can simulate a PBC on n qubits in a time 2 αn polyðnÞ, where α ≈ 0.94. This improves upon the brute-force simulation method, which takes time 2 n polyðnÞ. Our algorithm exploits the fact that n-fold tensor products of magic states admit a low-rank decomposition into n-qubit stabilizer states.

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Section: Appendix B: Stabilizer Decompositionsmentioning

confidence: 99%

Trading Classical and Quantum Computational Resources

2016

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“…[7][8][9][10][11]), the vast majority of design methods does not consider this metric. As an example in [7,11], a cycle representation was chosen and input cycles where partitioned into three subsets.…”

Section: Introductionmentioning

confidence: 99%

“…However, their approach makes use of a special class of templates. Finally, the work presented in [10] describes an exhaustive algorithm aiming to find a minimal depth quantum circuit using a special gate library. However, due to its exponential time complexity, it is only applicable to circuits with a small number of qubits.…”

Section: Introductionmentioning

confidence: 99%

Reducing the Depth of Quantum Circuits Using Additional Circuit Lines

Abdessaied

Wille

Soeken

et al. 2013

Reversible Computation

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Abstract. The synthesis of Boolean functions, as they are found in many quantum algorithms, is usually conducted in two steps. First, the function is realized in terms of a reversible circuit followed by a mapping into a corresponding quantum realization. During this process, the number of lines and the quantum costs of the resulting circuits have mainly been considered as optimization objectives thus far. However, beyond that also the depth of a quantum circuit is vital. Although first synthesis approaches that consider depth have recently been introduced, the majority of design methods did not consider this metric. In this paper, we introduce an optimization approach aiming for the reduction of depth in the process of mapping a reversible circuit into a quantum circuit. For this purpose, we present an improved (local) mapping of single gates as well as a (global) optimization scheme considering the whole circuit. In both cases, we incorporate the idea of exploiting additional circuit lines which are used in order to split a chain of serial gates. Our optimization techniques enable a concurrent application of gates which significantly reduces the depth of the circuit. Experiments show that reductions of approx. 40% on average can be achieved when following this scheme.

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“…The classical complexity of constructing a quantum circuit implementing |v is in O(k). In the controlled version of this circuit the number of gates remains O(k) ( [9], Theorem 1). In summary, we need O(log(1/ε)) gates to achieve precision ε.…”

Section: Precision and Complexity Analysismentioning

confidence: 99%

“…The original algorithm proposed in [8] uses a decomposition into single and two level unitaries. Each single and two level unitary may have a relatively large (yet, resulting in a blow up by at most a constant factor, [9]) implementation cost. An example is given by the CNOT gate, whose controlled version, the Toffoli gate, requires a strictly positive number of T gates, whereas none are needed for constructing the CNOT itself.…”

Section: Future Workmentioning

confidence: 99%

Asymptotically Optimal Approximation of Single Qubit Unitaries by Clifford andTCircuits Using a Constant Number of Ancillary Qubits

2013

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We present an algorithm for building a circuit that approximates single qubit unitaries with precision ε using O(log(1/ε)) Clifford and T gates and employing up to two ancillary qubits. The algorithm for computing our approximating circuit requires an average of O(log 2 (1/ε) log log(1/ε)) operations. We prove that the number of gates in our circuit saturates the lower bound on the number of gates required in the scenario when a constant number of ancillae are supplied, and as such, our circuits are asymptotically optimal. This results in significant improvement over the current state of the art for finding an approximation of a unitary, including the Solovay-Kitaev algorithm that requires O(log 3+δ (1/ε)) gates and does not use ancillae and the phase kickback approach that requires O(log 2 (1/ε) log log(1/ε)) gates, but uses O(log 2 (1/ε)) ancillae.

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A Meet-in-the-Middle Algorithm for Fast Synthesis of Depth-Optimal Quantum Circuits

Cited by 433 publications

References 37 publications

Trading Classical and Quantum Computational Resources

Trading Classical and Quantum Computational Resources

Reducing the Depth of Quantum Circuits Using Additional Circuit Lines

Asymptotically Optimal Approximation of Single Qubit Unitaries by Clifford andTCircuits Using a Constant Number of Ancillary Qubits

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