Automated Mapping Methods for the IBM Transmon Devices Kaitlin N. Smith and Mitchell A. Thornton Quantum Informatics Research Group Southern Methodist University Dallas, TX, USA fknsmith, [email protected] Abstract— Quantum computing and quantum information pro- generic reroute algorithms capable of implementing multi- cessing devices are becoming a reality. As a direct result of qubit operations on uncoupled qubits for any architecture this new technology, there is a demand for methods that allow specified by a coupling map is of interest for a quantum the automated translation of quantum processing algorithms into forms consistent with emerging device libraries. Quantum logic synthesis tool. Mapping quantum logic transformations circuits in their original form may not always be compatible from one form to another is a tedious task if done manually with a technology platform, so they must be transformed into because it quickly becomes difficult and error-prone as the technology-dependent models to be executed on a real quantum number of inputs and outputs of a circuit increases. The machines. This synthesis procedure must consider operational need to perform technology-dependent mapping on quantum constraints of the physical quantum implementation such as the maximum number qubits, the available gates, and the algorithms creates a demand for an automated synthesis tool connectivity between qubits for multi-qubit operations. In this that allows generalized quantum circuits to be transformed work, algorithms that assist in technology-dependent quantum into designs that only include the limited library of quantum circuit mapping in the case of limited qubit connectivity will be transformations that are executable on a specific platform. Two explored. The IBM Q devices are the example target technology, techniques that can be implemented in a technology-depentent and the CNOT operation is the multi-qubit gate with limited configurations for implementation. quantum synthesis tool were developed: the adjacent SWAP qubit reroute algorithm and the connectivity tree qubit reroute I. INTRODUCTION algorithm. These algorithms were successfully applied to Quantum information processing (QIP) will likely cause a generate quantum circuits targeted for evaluation on different new era of technological revolution due to all of the promise IBM machines with unique coupling requirements. that quantum algorithms hold. Quantum algorithms have been This paper will proceed as follows: Section II will provide a proven theoretically to outperform classical methods in com- brief summary of important concepts in quantum computation puting tasks that require large datasets or numerous iterations, while Section III introduces and describes the IBM QCs. The but these algorithms can only be implemented when reliable reroute methods are presented in Section IV and conclusions quantum computers (QCs) that contain an adequate number and future work are found in Section V. of qubits exist. Many successful QC prototypes containing II. BACKGROUND ON QUANTUM COMPUTATION a modest number of qubits have been developed, but these devices have their differences with respect to what types of A. The Qubit quantum algorithms, or circuits, they can execute. Because of The unit of quantum information is the quantum bit, or the variety between QCs, even between machines built using qubit. A qubit models information as a linear combination the same base technology, methods must be developed that of basis states. For example, j0i and j1i are a set of or- allow arbitrary quantum circuits to be mapped to technology- thonormal basis states in Dirac notation that represent the T dependent models. Generalized quantum algorithms must be two-dimensional column vectors of j0i = 1 0 and j1i = T transformed into technology-dependent forms to be executed 0 1 , respectively. The qubit can theoretically represent an on a real quantum machine, and this transformation process infinite continuum of states while in superposition of the of must not only consider the available gates for the platform but basis states, j0i and j1i as shown in also the connectivity that exists between qubits for multi-qubit operations. jΨi = α j0i + β j1i : (1) The IBM Q machines are publically available transmon- based devices that allow users to experiment with custom In Eqn. 1, the variables α and β are complex numbers, c, quantum algorithms at the gate level. The IBM devices, how- that take the form of c = a + bi where i is an imaginary ever, are limited to unique connectivity maps for multi-qubit number that takes the value i2 = −1. The probability that jΨi interactions with the two-qubit CNOT, or controlled-NOT, is measured to be j0i is equal to jαj2 and the probability that operator. Because it is common for many physical quantum jΨi is measured to be j1i is equal to jβj2. Once measured, realizations to have limited multi-qubit connectivity, finding qubits collapse into basis states, losing superposition. 12 TABLE I Qubits can be physically realized with a variety of different COMMON SINGLE- AND MULTI-QUBIT QUANTUM OPERATORS physical entities. These realizations can be classified into two types: atomic-scale qubits and mesoscopic qubits. Atomic- Operator Symbol Transformation Matrix scale qubits generally use actual particles, like atoms or ions, 0 1 and also include quasi-particles such as anyons, quantum dots, Pauli-X (NOT) 1 0 and Ni vacancies in diamond. The second class of qubits, the 0 −i mesoscopic forms, are realized as macroscopic circuits present Pauli-Y (Y) in an integrated circuit (IC), yet they exhibit microscopic i 0 quantum properties at supercooled temperatures. Mesoscopic 1 0 Pauli-Z (Z) qubits are generally classified according to the information- 0 −1 carrying medium in the circuit such as charge, flux, and phase. Quantum phenomena in mesoscopic qubits require extremely 1 1 Hadamard (H) p1 low electrical resistance for viability and are realized in the 2 1 −1 form of electrical circuits on ICs. These devices are often referred to as superconducting solid-state qubits. 1 0 Phase (S) Accidental measurement of a qubit’s superposition causes 0 i it to collapse into a basis state. This collapse, usually caused 1 0 π=8 (T) by an unintended interaction between the qubit and its envi- 0 eiπ=4 ronment, is called decoherence [1]. Some quantum systems 2 3 are more resilient to decoherence as compared to others, and 1 0 0 0 0 1 0 0 CNOT 6 7 in QIP circuit and QC design, there is a trade-off between 40 0 0 15 average decoherence times and the ability for qubits to interact. 0 0 1 0 Multi-qubit coupling is necessary to enable required operations 21 0 0 03 0 0 1 0 such as the CNOT or controlled-phase, but the ability for SWAP 6 7 40 1 0 05 qubits to interact with each other implies that qubits also 0 0 0 1 have the undesirable ability to interact with the environment, 21 0 0 0 0 0 0 03 increasing the probability of decoherence. Determining qubit 0 1 0 0 0 0 0 0 6 7 realizations that permit easy interaction with one another while 60 0 1 0 0 0 0 07 6 7 60 0 0 1 0 0 0 07 also resisting environmental coupling is currently an active Toffoli 6 7 60 0 0 0 1 0 0 07 area of research that accounts for, in part, the lack of a wide 6 7 60 0 0 0 0 1 0 07 consensus on the preferred implementation of a qubit. Many 40 0 0 0 0 0 0 15 current QC/QIP realization efforts are focused upon the class 0 0 0 0 0 0 1 0 of superconducting solid-state qubits as they are considered by many to offer the best tradeoff between qubit interaction versus average time of decoherence. U jΨini = jΨouti (2) B. Quantum Operators must be evaluated. The individual qubit input values are Quantum operators are implemented to purposely transform combined via tensor product to create the quantum state for qubit state so that meaningful quantum computation can occur. jΨini. Consider the quantum circuit pictured in Fig. 1. In this If a quantum algorithm is modeled as a circuit, quantum oper- circuit, two qubits, jai and jbi, are each represented by a ators can be viewed as quantum logic gates. These operators single horizontal line, but these horizontal lines should not are represented by a unique, unitary transformation matrix, be confused with conductors like those in in electrical circuit U. Common single- and multi-qubit operations are featured schematics. These qubits travel from left to right as time in Table I. progresses, passing though a CNOT gate. Together at the input Single quantum operators are combined to form quantum they form the value of jΨini which is found to be circuits, and quantum circuits can be described in a variety ways. Some of the most popular techniques include drawing jΨ i = ja i ⊗ jb i = j1i ⊗ j0i the circuits as graphs, as seen in Fig. 1, or describing them in in in with text in Quantum Assembly Language (QASM). 203 Understanding how information evolves in a quantum circuit 0 1 607 requires knowledge of matrix multiplication and tensor prod- jΨini = ⊗ = 6 7 = j10i : 1 0 415 ucts. As seen in Table I, quantum operators are represented by 0 transformation matrices. To determine the resulting quantum state, jΨouti, after after the input state, jΨini, undergoes a Determining the value of jΨouti requires Eqn. 2 to be utilized. gate transformation, U, the expression The transformation matrix of CNOT will take the place of the 13 generalized transformation matrix U in the equation, and the • ibmqx2 - 0: [1, 2], 1: [2], 3: [2, 4], 4: [2] equation will be evaluated as • ibmqx3 - 0:[1], 1: [2], 2: [3], 3: [14], 4: [3,5], 6: [7,11], 7: [10], 8: [7], 9: [8,10], 11: [10], 12: [5,11,13], 13: [4,14], jΨouti = U jΨini = CNOT j10i 15: [0,14] • ibmqx4 - 1: [0], 2: [0,1,4], 3: [2,4] 21 0 0 03 203 203 • ibmqx5 - 1:[2], 2:[3], 3:[4,14], 5:[4], 6:[7,11], 7:[10], 60 1 0 07 607 607 jΨouti = 6 7 6 7 = 6 7 = j11i : 8:[7], 9:[8,10], 11:[10], 12:[5,11,13], 13:[4,14], 40 0 0 15 415 405 15:[0,2,14] 0 0 1 0 0 1 The IBM simulator can host a maximum 20 qubits that are Finding the overall transfer function, U, that describes a unrestricted by a coupling map.