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On the realistic worst case analysis of quantum arithmetic circuits

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 Added by Alexandru Paler
 Publication date 2021
and research's language is English




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We provide evidence that commonly held intuitions when designing quantum circuits can be misleading. In particular we show that: a) reducing the T-count can increase the total depth; b) it may be beneficial to trade CNOTs for measurements in NISQ circuits; c) measurement-based uncomputation of relative phase Toffoli ancillae can make up to 30% of a circuits depth; d) area and volume cost metrics can misreport the resource analysis. Our findings assume that qubits are and will remain a very scarce resource. The results are applicable for both NISQ and QECC protected circuits. Our method uses multiple ways of decomposing Toffoli gates into Clifford+T gates. We illustrate our method on addition and multiplication circuits using ripple-carry. As a byproduct result we show systematically that for a practically significant range of circuit widths, ripple-carry addition circuits are more resource efficient than the carry-lookahead addition ones. The methods and circuits were implemented in the open-source QUANTIFY software.



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We present some basic integer arithmetic quantum circuits, such as adders and multipliers-accumulators of various forms, as well as diagonal operators, which operate on multilevel qudits. The integers to be processed are represented in an alternative basis after they have been Fourier transformed. Several arithmetic circuits operating on Fourier transformed integers have appeared in the literature for two level qubits. Here we extend these techniques on multilevel qudits, as they may offer some advantages relative to qubits implementations. The arithmetic circuits presented can be used as basic building blocks for higher level algorithms such as quantum phase estimation, quantum simulation, quantum optimization etc., but they can also be used in the implementation of a quantum fractional Fourier transform as it is shown in a companion work presented separately.
Quantum resource analysis is crucial for designing quantum circuits as well as assessing the viability of arbitrary (error-corrected) quantum computations. To this end, we introduce QUANTIFY, which is an open-source framework for the quantitative analysis of quantum circuits. It is based on Google Cirq and is developed with Clifford+T circuits in mind, and it includes the necessary methods to handle Toffoli+H and more generalised controlled quantum gates, too. Key features of QUANTIFY include: (1) analysis and optimisation methods which are compatible with the surface code, (2) choice between different automated (mixed polarity) Toffoli gate decompositions, (3) semi-automatic quantum circuit rewriting and quantum gate insertion methods that take into account known gate commutation rules, and (4) novel optimiser types that can be combined with different verification methods (e.g. truth table or circuit invariants like number of wires). For benchmarking purposes QUANTIFY includes quantum memory and quantum arithmetic circuits. Experimental results show that the frameworks performance scales to circuits with thousands of qubits.
146 - Maksim Levental 2021
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