There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

The Gottesman-Knill theorem says that a stabilizer circuit\char22{}that is, a quantum circuit consisting solely of controlled-NOT (CNOT), Hadamard, and phase gates\char22{}can be simulated efficiently on a classical computer. This paper improves that theorem in several directions. First, by removing the need for Gaussian elimination, we make the simulation algorithm much faster at the cost of a factor of 2 increase in the number of bits needed to represent a state. We have implemented the improved algorithm in a freely available program called CHP (CNOT-Hadamard-phase), which can handle thousands of qubits easily. Second, we show that the problem of simulating stabilizer circuits is complete for the classical complexity class $\ensuremath{\bigoplus}\mathsf{L}$, which means that stabilizer circuits are probably not even universal for classical computation. Third, we give efficient algorithms for computing the inner product between two stabilizer states, putting any $n$-qubit stabilizer circuit into a ``canonical form'' that requires at most $O({n}^{2}∕\mathrm{log}\phantom{\rule{0.2em}{0ex}}n)$ gates, and other useful tasks. Fourth, we extend our simulation algorithm to circuits acting on mixed states, circuits containing a limited number of nonstabilizer gates, and circuits acting on general tensor-product initial states but containing only a limited number of measurements.

https://www.scottaaronson.com/papers/chp6.pdf

Improved simulation of stabilizer circuits

The pressure of fundamental limits on classical computation and the promise of exponential speedups from quantum effects have recently brought quantum circuits (Proc. R. Soc. Lond. A, Math. Phys. Sci., vol. 425, p. 73, 1989) to the attention of the electronic design automation community (Proc. 40th ACM/IEEE Design Automation Conf., 2003), (Phys. Rev. A, At. Mol. Opt. Phy., vol. 68, p. 012318, 2003), (Proc. 41st Design Automation Conf., 2004), (Proc. 39th Design Automation Conf., 2002), (Proc. Design, Automation, and Test Eur., 2004), (Phys. Rev. A, At. Mol. Opt. Phy., vol. 69, p. 062321, 2004), (IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst., vol. 22, p. 710, 2003). Efficient quantum-logic circuits that perform two tasks are discussed: 1) implementing generic quantum computations, and 2) initializing quantum registers. In contrast to conventional computing, the latter task is nontrivial because the state space of an n-qubit register is not finite and contains exponential superpositions of classical bitstrings. The proposed circuits are asymptotically optimal for respective tasks and improve earlier published results by at least a factor of 2. The circuits for generic quantum computation constructed by the algorithms are the most efficient known today in terms of the number of most expensive gates [quantum controlled-NOTs (CNOTs)]. They are based on an analog of the Shannon decomposition of Boolean functions and a new circuit block, called quantum multiplexor (QMUX), which generalizes several known constructions. A theoretical lower bound implies that the circuits cannot be improved by more than a factor of 2. It is additionally shown how to accommodate the severe architectural limitation of using only nearest neighbor gates, which is representative of current implementation technologies. This increases the number of gates by almost an order of magnitude, but preserves the asymptotic optimality of gate counts

/pdf/synthesis-of-quantum-logic-circuits-7tglmmfb6j.pdf

Synthesis of quantum-logic circuits

We introduce a single-number metric, quantum volume, that can be measured using a concrete protocol on near-term quantum computers of modest size ($n\ensuremath{\lesssim}50$), and measure it on several state-of-the-art transmon devices, finding values as high as 16. The quantum volume is linked to system error rates, and is empirically reduced by uncontrolled interactions within the system. It quantifies the largest random circuit of equal width and depth that the computer successfully implements. Quantum computing systems with high-fidelity operations, high connectivity, large calibrated gate sets, and circuit rewriting toolchains are expected to have higher quantum volumes. The quantum volume is a pragmatic way to measure and compare progress toward improved system-wide gate error rates for near-term quantum computation and error-correction experiments.

Validating quantum computers using randomized model circuits

A digital combinational logic circuit is reversible if it maps each input pattern to a unique output pattern. Such circuits are of interest in quantum computing, optical computing, nanotechnology and low-power CMOS design. Synthesis approaches are not well developed for reversible circuits even for small numbers of inputs and outputs.In this paper, a transformation based algorithm for the synthesis of such a reversible circuit in terms of n × n Toffoli gates is presented. Initially, a circuit is constructed by a single pass through the specification with minimal look-ahead and no back-tracking. Reduction rules are then applied by simple template matching. The method produces near-optimal results for 3-input circuits and also produces very good results for larger problems.

/pdf/a-transformation-based-algorithm-for-reversible-logic-27g0s28056.pdf

A transformation based algorithm for reversible logic synthesis

Reversible or information-lossless circuits have applications in digital signal processing, communication, computer graphics, and cryptography. They are also a fundamental requirement in the emerging field of quantum computation. We investigate the synthesis of reversible circuits that employ a minimum number of gates and contain no redundant input-output line-pairs (temporary storage channels). We prove constructively that every even permutation can be implemented without temporary storage using NOT, CNOT, and TOFFOLI gates. We describe an algorithm for the synthesis of optimal circuits and study the reversible functions on three wires, reporting the distribution of circuit sizes. We also study canonical circuit decompositions where gates of the same kind are grouped together. Finally, in an application important to quantum computing, we synthesize oracle circuits for Grover's search algorithm, and show a significant improvement over a previously proposed synthesis algorithm.

/pdf/synthesis-of-reversible-logic-circuits-3j3y79pcmo.pdf

Synthesis of reversible logic circuits

We give quantum circuits that simulate an arbitrary two-qubit unitary operator up to a global phase. For several quantum gate libraries we prove that gate counts are optimal in the worst and average cases. Our lower and upper bounds compare favorably to previously published results. Temporary storage is not used because it tends to be expensive in physical implementations. For each gate library, the best gate counts can be achieved by a single universal circuit. To compute the gate parameters in universal circuits, we use only closed-form algebraic expressions, and in particular do not rely on matrix exponentials. Our algorithm has been coded in $\mathrm{C}++$.

https://arxiv.org/pdf/quant-ph/0308033.pdf

Minimal universal two-qubit controlled- NOT -based circuits

The three-input TOFFOLI gate is the workhorse of circuit synthesis for classical logic oper-ations on quantum data, e.g., reversible arithmetic circuits. In physical implementations,however, TOFFOLI gates are decomposed into six CNOT gates and several one-qubit gates.Though this decomposition has been known for at least 10 years, we provide here thefirst demonstration of its CNOT-optimality. We study three-qubit circuits which containless than six CNOT gates and implement a block-diagonal operator, then show that theyimplicitly describe the cosine-sine decomposition of a related operator. Leveraging thecanonical nature of such decompositions to limit one-qubit gates appearing in respectivecircuits, we prove that the n-qubit analogue of the TOFFOLI requires at least 2n CNOTgates. Additionally, our results offer a complete classification of three-qubit diagonaloperators by their CNOT-cost, which holds even if ancilla qubits are available.

On the CNOT-cost of TOFFOLI gates

Reversible or information-lossless circuits have applications in digital signal processing, communication, computer graphics and cryptography. They are also a fundamental requirement in the emerging field of quantum computation. We investigate the synthesis of reversible circuits that employ a minimum number of gates and contain no redundant input-output line-pairs (temporary storage channels). We prove constructively that every even permutation can be implemented without temporary storage using NOT, CNOT and TOFFOLI gates. We describe an algorithm for the synthesis of optimal circuits and study the reversible functions on three wires, reporting distributions of circuit sizes. Finally, in an application important to quantum computing, we synthesize oracle circuits for Grover's search algorithm, and show a significant improvement over a previously proposed synthesis algorithm.

/pdf/reversible-logic-circuit-synthesis-1lrui8vdcs.pdf

Vivek Shende

Papers

Synthesis of quantum-logic circuits

Synthesis of reversible logic circuits

Minimal universal two-qubit controlled- NOT -based circuits

On the CNOT-cost of TOFFOLI gates

Reversible logic circuit synthesis