Theme Article: Top Picks Extending the Frontier of Quantum Computers With Qutrits Pranav Gokhale, Jonathan M. Baker, Kenneth R. Brown Casey Duckering, and Frederic T. Chong Duke University University of Chicago Natalie C. Brown Georgia Institute of Technology Abstract—We advocate for a fundamentally different way to perform quantum computation by using three-level qutrits instead of qubits. In particular, we substantially reduce the resource requirements of quantum computations by exploiting a third state for temporary variables (ancilla) in quantum circuits. Past work with qutrits has demonstrated only constant factor improvements, owing to the log2(3) binary-to-ternary compression factor. We present a novel technique using qutrits to achieve a logarithmic runtime decomposition of the Generalized Toffoli gate using no ancilla—an exponential improvement over the best qubit-only equivalent. Our approach features a 70x improvement in total two-qudit gate count over the qubit-only decomposition. This results in improvements for important algorithms for arithmetic and QRAM. Simulation results under realistic noise models indicate over 90% mean reliability (fidelity) for our circuit, versus under 30% for the qubit- only baseline. These results suggest that qutrits offer a promising path toward extending the frontier of quantum computers. & RECENT ADVANCES IN both hardware and outcomes. In the coming years, we expect quan- software for quantum computation have demon- tum computing will have important applications strated significant progress toward practical in fields ranging from machine learning and optimization to drug discovery. While early research efforts focused on longer term systems Digital Object Identifier 10.1109/MM.2020.2985976 employing full error correction to execute large Date of publication 16 April 2020; date of current version 22 instances of algorithms like Shor factoring and May 2020. Grover search, recent work has focused on noisy 64 0272-1732 ß 2020 IEEE Published by the IEEE Computer Society IEEE Micro Authorized licensed use limited to: UNIV OF CHICAGO LIBRARY. Downloaded on September 03,2020 at 13:54:21 UTC from IEEE Xplore. Restrictions apply. intermediate scale quantum (NISQ) computa- The net result of our work is to extend the fron- tion. The NISQ regime considers near-term tier of what quantum computers can compute. In machines with just tens to hundreds of quantum particular, the frontier is defined by the zone in bits (qubits) and moderate errors. which every machine qubit is a data qubit, for Given the severe constraints on example, a 100-qubit algorithm quantum resources, it is critical to Given the severe running on a 100-qubit machine. fully optimize the compilation of a constraints on quantum This is indicated by the yellow quantum algorithm in order to have resources, it is critical region in Figure 1. In this frontier successful computation. Prior archi- to fully optimize the zone, we do not have room for tectural research has explored tech- compilation of a nondata workspace qubits known niques such as mapping, scheduling, quantum algorithm as ancilla. The lack of ancilla in and parallelism to extend the amount in order to have the frontier zone is a costly con- of useful computation possible. In successful straint that generally leads to inef- this article, we consider another computation. ficient circuits. For this reason, technique: quantum trits (qutrits). typical circuits instead operate While quantum computation is typically below the frontier zone, with many machine expressed as a two-level binary abstraction of qubits used as ancilla. This article demonstrates qubits, the underlying physics of quantum that ancilla can be substituted with qutrits, systems are not intrinsically binary. Whereas enabling us to extend the ancilla-free frontier zone classical computers operate in binary states of quantum computation. at the physical level (e.g., clipping above and below a threshold voltage), quantum com- puters have natural access to an infinite BACKGROUND spectrum of discrete energy levels. In fact, A qubit is the fundamental unit of quantum hardware must actively suppress higher level computation. Compared to their classical coun- states in order to achieve the two-level qubit terparts which take values of either 0 and 1, approximation. Hence, using three-level qut- qubits may exist in a superposition of the two rits is simply a choice of including an addi- states. We designate these two basis states as 0 j i tional discrete energy level, albeit at the cost and 1 and can represent any qubit as j i of more opportunities for error. c a 0 b 1 with a 2 b 2 1. a 2 and j i¼ j iþ j i k k þk k ¼ k k Prior work on qutrits (or more generally, b 2 correspond to the probabilities of measur- k k d-level qudits) identified only constant factor ing 0 and 1 , respectively. j i j i gains from extending beyond qubits. In Quantum states can be acted on by quan- general, this prior work1 has emphasized the tum gates, which preserve valid probability information compression advantages of qut- distributions that sum to 1 and guarantee rits. For example, N qubits can be expressed reversibility. For example, the X gate trans- as N=log2 3 qutrits, which leads to forms a state c a 0 b 1 to X c ð Þ j i¼ j iþ j i j i¼ log2 3 1:6 constant factor improvements in b 0 a 1 . The X gate is also an example of a ð Þ j iþ j i runtimes. classical reversible operation, equivalent to the Our approach utilizes qutrits in a novel fash- NOT operation. In quantum computation, we have ion, essentially using the third state as tempo- a single irreversible operation called measurement rary storage, but at the cost of higher per- that transforms a quantum state into one of the operation error rates. Under this treatment, the two basis states with a given probability based on runtime (i.e., circuit depth or critical path) is a and b. asymptotically faster, and the reliability of com- In order to interact different qubits, two-qubit putations is also improved. Moreover, our operations are used. The CNOT gate appears both approach only applies qutrit operations in an in classical reversible computation and in quan- intermediary stage: The input and output are tum computation. It has a control qubit and a tar- still qubits, which is important for initialization get qubit. When the control qubit is in the 1 state, 2;3 j i and measurement on real devices. the CNOT performs a NOT operation on the target. May/June 2020 65 Authorized licensed use limited to: UNIV OF CHICAGO LIBRARY. Downloaded on September 03,2020 at 13:54:21 UTC from IEEE Xplore. Restrictions apply. Top Picks In a three-level system, we consider the computational basis states 0 , 1 , and 2 for j i j i j i qutrits. A qutrit state c may be represented j i analogously to a qubit as c a 0 b 1 j i¼ j iþ j iþ g 2 , where a2 b2 g2 1. Qutrits are manipu- j i þ þ ¼ lated in a similar manner to qubits; however, there are additional gates which may be per- formed on qutrits. For instance, in quantum binary logic, there is only a single X gate. In ternary, there are three X gates denoted X01, X02,andX12. Each of these Xij can be viewed as swapping i with j and leaving j i j i the third basis element unchanged. For example, for a qutrit c a 0 b 1 g 2 , applying X02 j i¼ j iþ j iþ j i produces X02 c g 0 b 1 a 2 .Thereare j i¼ j iþ j iþ j i Figure 1. Frontier of what quantum hardware can two additional nontrivial operations on a single execute is the yellow region adjacent to the 45 line. trit. They are X 1 and X 1 operations, which per- þ À In this region, each machine qubit is a data qubit. form the addition/subtraction modulo 3. Typical circuits rely on nondata ancilla qubits for Just as single qubit gates have qutrit analogs, workspace and therefore operate below the frontier. the same holds for two qutrit gates. For example, consider the CNOT operation, where an X gate is The cnot gate serves a special role in quantum performed conditioned on the control being in the computation, allowing quantum states to become 1 state. For qutrits, any of the X gates presented j i entangled so that a pair of qubits cannot be above may be performed, conditioned on the con- described as two individual qubit states. Any oper- trol being in any of the three possible basis states. ation may be conditioned on one or more controls. In order to evaluate a decomposition of a Many classical operations, such as AND and or quantum circuit, we consider quantum circuit gates, are irreversible and therefore cannot costs. The space cost of a circuit, i.e., the num- directly be executed as quantum gates. For ber of qubits (or qutrits), is referred to as circuit example, consider the output of 1 from an OR width. Requiring ancilla increases the circuit gate with two inputs. With only this information width and, therefore, the space cost of a circuit. about the output, the value of the inputs cannot The time cost for a circuit is the depth of a cir- be uniquely determined. These operations can cuit. The depth is given as the length of the criti- be made reversible by the addition of extra, tem- cal path from input to output. porary ancilla bits initialized to 0 . j i Physical systems in classical hardware are typi- cally binary.