ISCA 2019 Submission #576 Confidential Draft: DO NOT DISTRIBUTE Asymptotic Improvements to Quantum Circuits via Qutrits ABSTRACT energy levels. In fact, hardware must actively suppress higher Quantum computation is traditionally expressed in terms of level states in order to achieve the two-level qubit approxi- quantum bits, or qubits. In this work, we instead consider mation. Hence, using three-level qutrits is simply a choice three-level qutrits. Past work with qutrits has demonstrated of including an additional discrete energy level, albeit at the cost of more opportunities for error. only constant factor improvements, owing to the log2(3) binary-to-ternary compression factor. We present a novel Prior work on qutrits (or more generally, d-level qudits) technique using qutrits to achieve a logarithmic depth (run- identified only constant factor gains from extending beyond time) decomposition of the Generalized Toffoli gate using qubits. In general, this prior work [9] has emphasized the no ancilla–a significant improvement over linear depth for information compression advantages of qutrits. For example, N qubits can be expressed as N qutrits, which leads to the best qubit-only equivalent. Our circuit construction also log2(3) features a 70x improvement in two-qudit gate count over log2(3) ≈ 1:6-constant factor improvements in runtimes. the qubit-only equivalent decomposition. This results in cir- Our approach utilizes qutrits in a novel fashion, essentially cuit cost reductions for important algorithms like quantum using the third state as temporary storage, but at the cost of neurons and Grover search. We develop an open-source cir- higher per-operation error rates. Under this treatment, the cuit simulator for qutrits, along with realistic near-term noise runtime (i.e. circuit depth or critical path) is asymptotically models which account for the cost of operating qutrits. Simu- faster, and the reliability of computations is also improved. lation results for these noise models indicate over 90% mean Moreover, our approach only applies qutrit operations in reliability (fidelity) for our circuit construction, versus un- an intermediary stage: the input and output are still qubits, der 30% for the qubit-only baseline. These results suggest which is important for initialization and measurement on real that qutrits offer a promising path towards scaling quantum devices [10, 11]. computation. 1. INTRODUCTION Infeasible, Recent advances in both hardware and software for quan- not enough qubits tum computation have demonstrated significant progress to- wards practical outcomes. In the coming years, we expect quantum computing will have important applications in fields ranging from machine learning and optimization [1] to drug discovery [2]. While early research efforts focused on longer- term systems employing full error correction to execute large instances of algorithms like Shor factoring [3] and Grover search [4], recent work has focused on NISQ (Noisy Inter- Number of Data Qubits Typical mediate Scale Quantum) computation [5]. The NISQ regime Frontier, no space for ancillas considers near-term machines with just tens to hundreds of quantum bits (qubits) and moderate errors. Feasible, Given the severe constraints on quantum resources, it is can use ancillas critical to fully optimize the compilation of a quantum algo- rithm in order to have successful computation. Prior archi- Number of Qubits on Machine tectural research has explored techniques such as mapping, scheduling, and parallelism [6,7,8] to extend the amount Figure 1: The frontier of what quantum hardware can of useful computation possible. In this work, we consider execute is the yellow region adjacent to the 45° line. In another technique: quantum trits (qutrits). this region, each machine qubit is a data qubit. Typi- While quantum computation is typically expressed as a cal circuits rely on non-data ancilla qubits for workspace two-level binary abstraction of qubits, the underlying physics and therefore operate below the frontier. of quantum systems are not intrinsically binary. Whereas classical computers operate in binary states at the physical The net result of our work is to extend the frontier of what level (e.g. clipping above and below 2.5V), quantum com- quantum computers can compute. In particular, the frontier puters have natural access to an infinite spectrum of discrete is defined by the zone in which every machine qubit is a 1 data qubit, for example a 100-qubit algorithm running on a computation and in quantum computation. It has a control 100-qubit machine. This is indicated by the yellow region in qubit and a target qubit. When the control qubit is in the j1i Figure1. In this frontier zone, we do not have room for non- state, the CNOT performs a NOT operation on the target. The data workspace qubits known as ancilla. The lack of ancilla CNOT gate serves a special role in quantum computation, in the frontier zone is a costly constraint that generally leads allowing quantum states to become entangled so that a pair to inefficient circuits. For this reason, typical circuits instead of qubits cannot be described as two individual qubit states. operate below the frontier zone, with many machine qubits Any operation may be conditioned on one or more controls. used as ancillas. Our work demonstrates that ancillas can Many classical operations, such as AND and OR gates, be substituted with qutrits, enabling us to operate efficiently are irreversible and therefore cannot directly be executed as within the ancilla-free frontier zone. quantum gates. For example, consider the output of 1 from We highlight the three primary contributions of our work: an OR gate with two inputs. With only this information about the output, the value of the inputs cannot be uniquely 1. A circuit construction based on qutrits that leads to determined. These operations can be made reversible by the asymptotically faster circuits (633N ! 38log N) than 2 addition of extra, temporary workspace bits initialized to 0. equivalent qubit-only constructions. We also reduce Using a single additional ancilla, the AND operation can be total gate counts from 397N to 6N. computed reversibly as in Figure2. 2. An open-source simulator, based on Google’s Cirq [12], which supports realistic noise simulation for qutrit (and jq i • jq i qudit) circuits. 0 0 jq1i • jq1i 3. Simulation results, under realistic noise models, which demonstrate our circuit construction outperforms equiv- j0i jq0 AND q1i alent qubit circuits in terms of error. For our bench- marked circuits, our reliability advantage ranges from Figure 2: Reversible AND circuit using a single ancilla 2x for trapped ion noise models up to more than 10,000x bit. The inputs are on the left, and time flows rightward for superconducting noise models. For completeness, to the outputs. This AND gate is implemented using a we also benchmark our circuit against a qubit-only con- Toffoli (CCNOT) gate with inputs q0, q1 and a single an- struction augmented by an ancilla and find our construc- cilla initialized to 0. At the end of the circuit, q0 and q1 tion is still more reliable. are preserved, and the ancilla bit is set to 1 if and only if both other inputs are 1. The rest of this paper is organized as follows: Section2 presents relevant background about quantum computation Physical systems in classical hardware are typically binary. and Section3 outlines related prior work that we benchmark However, in common quantum hardware, such as in super- our work against. Section4 demonstrates our key circuit conducting and trapped ion computers, there is an infinite construction, and Section5 surveys applications of this con- spectrum of discrete energy levels. The qubit abstraction is struction toward important quantum algorithms. Section6 an artificial approximation achieved by suppressing all but introduces our open-source qudit circuit simulator. Section7 the lowest two energy levels. Instead, the hardware may be explains our noise modeling methodology (with full details configured to manipulate the lowest three energy levels by in AppendixA), and Section8 presents simulation results for operating on qutrits. In general, such a computer could be these noise models. Finally, we discuss our results at a higher configured to operate on any number of d levels, except as d level in Section9. increases the number of opportunities for error, termed error channels, increases. Here, we focus on d = 3 with which we 2. BACKGROUND achieve the desired improvements to the Generalized Toffoli A qubit is the fundamental unit of quantum computation. gate. Compared to their classical counterparts which take values In a three level system, we consider the computational of either 0 and 1, qubits may exist in a superposition of basis states j0i, j1i, and j2i for qutrits. A qutrit state jyi may the two states. We designate these two basis states as j0i be represented analogously to a qubit as jyi = a j0i+b j1i+ and j1i and can represent any qubit as jyi = a j0i + b j1i g j2i, wherekak2 +kbk2 +kgk2 = 1. Qutrits are manipulated with kak2 + kbk2 = 1. kak2 and kbk2 correspond to the in a similar manner to qubits; however, there are additional probabilities of measuring j0i and j1i respectively. gates which may be performed on qutrits. Quantum states can be acted on by quantum gates which (a) For instance, in quantum binary logic, there is only a sin- preserve valid probability distributions that sum to 1 and (b) gle X gate. In ternary, there are three X gates denoted X01, guarantee reversibility. For example, the X gate transforms X02, and X12. Each of these Xi j for i 6= j can be viewed as a state jyi = a j0i + b j1i to X jyi = b j0i + a j1i. The X swapping jii with j ji and leaving the third basis element un- gate is also an example of a classical reversible operation, changed.