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International Symposium on Embedded Computing and System Design (ISED)

Efficient Realizations of CNOT gates in IBM’s Quantum Computers

Alexandre A. A. de Almeida, Gerhard W. Dueck, Alexandre César Rodrigues da Silva

Published Dec 1, 2018• 6 citations
Publisher

Abstract

IBM's quantum computers implement gates from Clifford +T gate library. All single qubit gates are implemented, but only a subset of the possible CNOT are provided. It is well known that the functionally of the missing gates can be achieved by a sequence of gates. The sequence of gates is based on SWAP gates. Up to seven elementary gates are required to implement a SWAP gate. In this paper we show how the same effect can be achieved with fewer gates. To show the potential of the proposed transformations, an example is presented where a reduction of 44% in the gate count and a 26% reduction in the number of levels for IBM's QX5 computer is achieved. An algorithm that is considered state of the art, is used for the comparison.

DOI
10.1109/ised.2018.8704098

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IBM's quantum computers implement gates from Clifford +T gate library. All single qubit gates are implemented, but only a subset of the possible CNOT are provided. It is well known that the functionally of the missing gates can be achieved by a sequence of gates. The sequence of gates is based on SWAP gates. Up to seven elementary gates are required to implement a SWAP gate. In this paper we show how the same effect can be achieved with fewer gates. To show the potential of the proposed transformations, an example is presented where a reduction of 44% in the gate count and a 26% reduction in the number of levels for IBM's QX5 computer is achieved. An algorithm that is considered state of the art, is used for the comparison.
IBM architectures impose some restrictions the quantum circuits that can be implemented. Only gates from Clifford+T gate library can be used and not all of the CNOT gates are available. Some of the CNOT need to be mapped into a sequence of gates. In this paper we present a set of mappings to conform to the restrictions imposed by IBM's architectures. These mappings require fewer gates than SWAP gates. It is well known, that permuting the qubits will yield circuits with different number of gates. The design in this paper uses efficient mappings with qubit permutations to obtain circuits with a reduced number of gates. Results have shown that the proposed approach reduces circuits by up to 64% compared with Qiskit and up to 42% compared with another mapping algorithm.
IBM has made several quantum computers available to researchers around the world via cloud services. Two architectures with five qubits, one with 16, and one with 20 qubits are available to run experiments. The IBM architectures implement gates from the Clifford+T gate library. However, each architecture only implements a subset of the possible CNOT gates. In this paper, we show how Clifford+T circuits can efficiently be mapped into the two IBM quantum computers with 5 qubits. We further present an algorithm and a set of circuit identities that may be used to optimize the Clifford+T circuits in terms of gate count and number of levels. It is further shown that the optimized circuits can considerably reduce the gate count and number of levels and thus produce results with better fidelity.
IBM offers quantum processors for Clifford+T circuits. The only restriction is that not all CNOT gates are implemented and must be substituted with alternate sequences of gates. Each CNOT has its own mapping with a respective cost. However, by permuting the qubits, the number of CNOT that need mappings can be reduced. The problem is to find a good permutation without an exhaustive search. In this paper we propose a solution for this problem. The permutation problem is formulated as an Integer Linear Programming (ILP) problem. Solving the ILP problem, the lowest cost permutation for the CNOT mappings is guaranteed. To test and validated the proposed formulation, quantum architectures with 5 and 16 qubits were used. The ILP formulation along with mapping techniques found circuits with up to 64% fewer gates than other approaches.
Currently, computing paradigm is ruled by the laws of classical physics and will continue till the dimensions of transistors reaches the size of atomic particles. Quantum computing (QC) is a new technology that employs laws of quantum mechanics to address issues such as irreversibility and power dissipation that are outside the scope of conventional computing models. QC offers a powerful stage for solving complex and compound problems like searching and number factoring. A critical task in utilizing quantum physics in many application fields is circuit design using reversible quantum gates. Unitary decomposition is a popular scheme for mapping quantum algorithms to any set of elementary quantum operations. Using decomposition techniques enables transformation of unitary matrices into fundamental quantum gates, which is critical for running algorithms on prevailing quantum computers. Any 3x3 reversible quantum gate can be decomposed into single-qubit rotation gates and two qubit CNOT gates. This paper is the first to present quantum implementations of FRSG1 and JTF1 gates into CNOT gates and single qubit U3 gates with different optimization levels on a platform provided by IBM. These gates are important in a variety of practical applications like Stochastic computing and parity generation circuits. In FRSG1 the implementation count of single qubit gates decreases by 56% and the count of two qubit gates by 15% after optimization, whereas in JTF1 gate the single qubit gate count reduces by 71% and the number of two qubit gates reduces by 41%.
While mapping a quantum circuit to the physical layer one has to consider the numerous constraints imposed by the underlying hardware architecture. Connectivity of the physical qubits is one such constraint that restricts two-qubit operations, such as CNOT, to “connected” qubits. SWAP gates can be used to place the logical qubits on admissible physical qubits, but they entail a significant increase in CNOT-count. In this article, we consider the problem of reducing the CNOT-count in Clifford+T circuits on connectivity-constrained architectures, like noisy intermediate-scale quantum (NISQ) computing devices. We “slice” the circuit at the position of Hadamard gates and “build” the intermediate{CNOT,T}\{\text {CNOT},{T}\}{CNOT,T}subcircuits using Steiner trees, significantly improving on previous methods. We compared the performance of our algorithms while mapping different benchmark and random circuits to some well-known architectures, such as 9-qubit square grid, 16-qubit square grid, Rigetti 16-qubit Aspen, 16-qubit IBM QX5, and 20-qubit IBM Tokyo. Our methods give less CNOT-count compared to Qiskit and TKET transpiler as well as using SWAP gates. Assuming most of the errors in an NISQ circuit implementation are due to CNOT errors, then our method would allow circuits with a few times more CNOT gates be reliably implemented than the previous methods would permit.
We study the implementation of quantum channels with quantum computers while minimizing the experimental cost, measured in terms of the number of controlled-not (cnot) gates required (single-qubit gates are free). We consider three different models. In the first, the quantum circuit model (QCM), we consider sequences of single-qubit and cnot gates and allow qubits to be traced out at the end of the gate sequence. In the second (RandomQCM), we also allow external classical randomness. In the third (MeasuredQCM) we also allow measurements followed by operations that are classically controlled on the outcomes. We prove lower bounds on the number of cnot gates required and give near-optimal decompositions in almost all cases. Our main result is a MeasuredQCM circuit for any channel from mmm qubits to nnn qubits that uses at most one ancilla and has a low cnot count. We give explicit examples for small numbers of qubits that provide the lowest known cnot counts.