Abstract: Among the embodiments disclosed herein are quantum circuits (and associated compilation techniques) for performing Shor's quantum algorithm to factor n-bit integers. Example embodiments of the circuits use only 2n+2 qubits. In contrast to previous space-optimized implementations, embodiments of the disclosed technology feature a purely Toffoli-based modular multiplication circuit. Certain other example modular multiplication circuits disclosed herein are based on an (in-place) constant-adder that uses dirty ancilla qubits to achieve a size in (n log n) and a depth in (n).

Abstract: Quantum circuits and associated methods use Repeat-Until-Success (RUS) circuits to perform approximate multiplication and approximate squaring of input values supplied as rotations encoded on ancilla qubits. So-called gearbox and programmable ancilla circuits are coupled to encode even or odd products of input values as a rotation of a target qubit. In other examples, quantum RUS circuits provide target qubit rotations that are associated with reciprocals using series expansion representations.

Abstract: Quantum algorithms to solve practical problems in quantum chemistry, materials science, and matrix inversion often involve a significant amount of arithmetic operations. These arithmetic operations are to be carried out in a way that is amenable to the underlying fault-tolerant gate set, leading to an optimization problem to come close to the Pareto-optimal front between number of qubits and overall circuit size. In this disclosure, a quantum circuit library is provided for floating-point addition and multiplication. Circuits are presented that are automatically generated from classical Verilog implementations using synthesis tools and compared with hand-generated and hand-optimized circuits. Example circuits were constructed and tested using the software tools LIQUi| and RevKit.

Abstract: Repeat-Until-Success (RUS) circuits are compiled in a Clifford+T basis by selecting a suitable cyclotomic integer approximation of a target rotation so that the rotation is approximated within a predetermined precision. The cyclotomic integer approximation is randomly modified until a modified value can be expanded into a single-qubit unitary matrix by solving one or more norm equations. The matrix is then expanded into a two-qubit unitary matrix of special form, which is then decomposed into an optimal two-qubit Clifford+T circuit. A two-qubit RUS circuit using a primary qubit and an ancillary qubit is then obtained based on the latter decomposition. An alternate embodiment is disclosed that keeps the total T-depth of the derived circuit small using at most 3 additional ancilla qubits. Arbitrary unitary matrices defined over the cyclotomic field of 8th roots of unity are implemented with RUS circuits.

Abstract: Ripple-carry and carry look-ahead adders for ternary addition and other operations include circuits that produce carry values or carry status indicators that can be stored on qutrit registers associated with input values to be processed. Inverse carry circuits are situated to reverse operations associated with the production of carry values or carry status indicators, and restored values are summed with corresponding carry values to produce ternary sums.

Abstract: In this application, example methods for performing quantum Montgomery arithmetic are disclosed. Additionally, circuit implementations are disclosed for reversible modular arithmetic, including modular addition, multiplication and inversion, as well as reversible elliptic curve point addition. This application also shows that elliptic curve discrete logarithms on an elliptic curve defined over an n-bit prime field can be computed on a quantum computer with at most 9n+2 ?log2(n)?+10 qubits using a quantum circuit of at most 512n3 log2(n)+3572n3 Toffoli gates.

Abstract: This application concerns methods, apparatus, and systems for performing quantum circuit synthesis and/or for implementing the synthesis results in a quantum computer system. In certain example embodiments: a universal gate set, a target unitary described by a target angle, and target precision is received (input); a corresponding quaternion approximation of the target unitary is determined; and a quantum circuit corresponding to the quaternion approximation is synthesized, the quantum circuit being over a single qubit gate set, the single qubit gate set being realizable by the given universal gate set for the target quantum computer architecture.

Abstract: The disclosed technology includes, among other innovations, a framework for resource efficient compilation of higher-level programs into lower-level reversible circuits. In particular embodiments, the disclosed technology reduces the memory footprint of a reversible network implemented in a quantum computer and generated from a higher-level program. Such a reduced-memory footprint is desirable in that it addresses the limited availability of qubits available in many target quantum computer architectures.

Abstract: Among the embodiments disclosed herein are quantum circuits (and associated compilation techniques) for performing Shor's quantum algorithm to factor n-bit integers. Example embodiments of the circuits use only 2n+2 qubits. In contrast to previous space-optimized implementations, embodiments of the disclosed technology feature a purely Toffoli-based modular multiplication circuit. Certain other example modular multiplication circuits disclosed herein are based on an (in-place) constant-adder that uses dirty ancilla qubits to achieve a size in (n log n) and a depth in (n).

Abstract: A Probabilistic Quantum Circuit with Fallback (PQFs) is composed as a series of circuit stages that are selected to implement a target unitary. A final stage is conditioned on unsuccessful results of all the preceding stages as indicated by measurement of one or more ancillary qubits. This final stage executes a fallback circuit that enforces deterministic execution of the target unitary at a relatively high cost (mitigated by very low probability of the fallback). Specific instances of general PQF synthesis method and are disclosed with reference to the specific Clifford+T, Clifford+V and Clifford+?/12 bases. The resulting circuits have expected cost in logb(1/?(log(log(1/?)))+const wherein b is specific to each basis. The three specific instances of the synthesis have polynomial compilation time guarantees.

Abstract: Quantum circuits and associated methods use Repeat-Until-Success (RUS) circuits to perform approximate multiplication and approximate squaring of input values supplied as rotations encoded on ancilla qubits. So-called gearbox and programmable ancilla circuits are coupled to encode even or odd products of input values as a rotation of a target qubit. In other examples, quantum RUS circuits provide target qubit rotations that are associated with reciprocals using series expansion representations.

Abstract: The generation of reversible circuits from high-level code is desirable in a variety of application domains, including low-power electronics and quantum computing. However, little effort has been spent on verifying the correctness of the results, an issue of particular importance in quantum computing where such circuits are run on all inputs simultaneously. Disclosed herein are example reversible circuit compilers as well as tools and techniques for verifying the compilers. Example compilers disclosed herein compile a high-level language into combinational reversible circuits having a reduced number of ancillary bits (ancilla bits) and further having provably clean temporary values.

Abstract: Repeat-Until-Success (RUS) circuits are compiled in a Clifford+T basis by selecting a suitable cyclotomic integer approximation of a target rotation so that the rotation is approximated within a predetermined precision. The cyclotomic integer approximation is randomly modified until a modified value can be expanded into a single-qubit unitary matrix by solving one or more norm equations. The matrix is then expanded into a two-qubit unitary matrix of special form, which is then decomposed into an optimal two-qubit Clifford+T circuit. A two-qubit RUS circuit using a primary qubit and an ancillary qubit is then obtained based on the latter decomposition. An alternate embodiment is disclosed that keeps the total T-depth of the derived circuit small using at most 3 additional ancilla qubits. Arbitrary unitary matrices defined over the cyclotomic field of 8th roots of unity are implemented with RUS circuits.

Abstract: Systems and methods for transforming an initial quantum state to a target quantum state are disclosed. The initial quantum state is denoted by superposed initial quantum sample states and the target quantum state is denoted by superposed target quantum sample states. The initial quantum state is initialized with a set of primary registers for the initial quantum state and with at least one ancillary register. The initial quantum state is transformed such that the set of primary registers reflects the initial quantum sample states and the at least one ancillary register is varied to compose an intermediate quantum state. In addition, the intermediate quantum state is amplified by implementing quantum state rotations in accordance with a plurality of reflections on the intermediate quantum state such that the reflections result in the target quantum sample states of the target quantum state with a discarding of the at least one ancillary register.

Abstract: Systems and methods to communicate securely includes communicating quantum encryption data on a first wavelength-division multiplexing passive optical network (WDM-PON); and communicating data over separate classical channels of a second WDM-PON, wherein the second WDM-PON synchronizes with the first WDM-PON while providing data communication over the classical channels.

Abstract: A quantum mechanical credit unit includes a plurality of qubit strings stored in computer readable storage media and configured for comparison with challenge questions during a verification procedure. The plurality of qubit strings is stored in at least k registers where k is a selected security number for the credit unit. An information register stores information about qubit strings that remain unused to provide the used qubit strings during the verification procedure. A unique serial number is configured to identify the credit unit without association with its holder or the qubit strings. Issuance and verification methods for the credit unit are also disclosed.

Abstract: A method includes performing quantum state tomography from the statistics of a collection of measurements, each of which has only two possible outcomes and has the feature of being a measurement of a single qubit. By carefully choosing the measurements it becomes possible to infer the state of a quantum system from the statistics. Moreover, the function which computes the state from the measurement statistics can be computed efficiently in the dimension of the underlying system. Reconstructing the quantum state is performed in accordance with the following expression: ? = ( 2 d ? ? i = 1 d 2 - 1 ? ? p i ? P i + ( 1 - p i ) ? ( 1 - P i ) ) - ( d 2 - 2 d ) ? I d , where d is the dimension of the quantum mechanical system, ? is the state of the quantum mechanical system, Id denotes the identity operator, Pi is one of the plurality of measurement projectors, and pi is the probability for the measurement projector Pi.

Abstract: Systems and methods for transforming an initial quantum state to a target quantum state are disclosed. The initial quantum state is denoted by superposed initial quantum sample states and the target quantum state is denoted by superposed target quantum sample states. The initial quantum state is initialized with a set of primary registers for the initial quantum state and with at least one ancillary register. The initial quantum state is transformed such that the set of primary registers reflects the initial quantum sample states and the at least one ancillary register is varied to compose an intermediate quantum state. In addition, the intermediate quantum state is amplified by implementing quantum state rotations in accordance with a plurality of reflections on the intermediate quantum state such that the reflections result in the target quantum sample states of the target quantum state with a discarding of the at least one ancillary register.

Abstract: A system and method for dynamical decoupling of a quantum system includes forming a graph including elements to account for decoupling sequence effects represented as nodes in the graph and soft pulses applied being represented as edges in the graph. Sequences which visit edges and nodes in the graph are provided. Binary strings corresponding to the nodes in a coordinate system are mapped using a fixed linear error correcting code. A decoupling method is provided based upon a matrix formed using the error correcting code to determine features of the soft pulses to decouple environmental effects from the quantum system.

Abstract: A quantum mechanical credit unit includes a plurality of qubit strings stored in computer readable storage media and configured for comparison with challenge questions during a verification procedure. The plurality of qubit strings is stored in at least k registers where k is a selected security number for the credit unit. An information register stores information about qubit strings that remain unused to provide the used qubit strings during the verification procedure. A unique serial number is configured to identify the credit unit without association with its holder or the qubit strings. Issuance and verification methods for the credit unit are also disclosed.