Abstract: A method of training a quantum computer employs quantum algorithms. The method comprises loading, into the quantum computer, a description of a quantum Boltzmann machine, and training the quantum Boltzmann machine according to a protocol, wherein a classification error is used as a metric for the protocol.

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: Embodiments of the disclosed technology employ parametric coordinate ascent to train a quantum circuit. In certain implementations, parameters (e.g., variational parameters) are learned by coordinate ascent using closed form equations. This strategy helps ensure monotonic convergence to local maxima in parameter space at predictable convergence rates and eliminates the overhead due to hyperparameter sweeps.

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/?)+O(log(log(1/?)))+const wherein b is specific to each basis. The three specific instances of the synthesis have polynomial compilation time guarantees.

Abstract: Certain ensembles of metapletic anyons allow for topologically protected encoding and processing of quantum information. Such processing is done by sequences of gates (circuits) drawn from a certain basis of unitary metaplectic gates. A subject unitary operator required for the desired processing can be approximated to any desired precision by a circuit that has to be effectively and efficiently synthesized on a classical computer. Synthesis methods use unitary reflection operators that can be represented either exactly or by ancilla-assisted approximation over the basis of metaplectic gates based on cost-optimizing determinations made by the synthesis algorithm.

Abstract: Embodiments of a new approach for training a class of quantum neural networks called quantum Boltzmann machines are disclosed. in particular examples, methods for supervised training of a quantum Boltzmann machine are disclosed using an ensemble of quantum states that the Boltzmann machine is trained to replicate. Unlike existing approaches to Boltzmann training, example embodiments as disclosed herein allow for supervised training even in cases where only quantum examples are known (and not probabilities from quantum measurements of a set of states). Further, this approach does not require the use of approximations such as the Golden-Thompson inequality.

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: Quantum circuits and circuit designs are based on factorizations of diagonal unitaries using a phase context. The cost/complexity of phase sparse/phase dense approximations is compared, and a suitable implementation is selected. For phase sparse implementations in the Clifford+T basis, required entangling circuits are defined based on a number of occurrences of a phase in the phase context in a factor of the diagonal unitary.

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: 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: Certain ensembles of metapletic anyons allow for topologically protected encoding and processing of quantum information. Such processing is done by sequences of gates (circuits) drawn from a certain basis of unitary metaplectic gates. A subject unitary operator required for the desired processing can be approximated to any desired precision by a circuit that has to be effectively and efficiently synthesized on a classical computer. Synthesis methods use unitary reflection operators that can be represented either exactly or by ancilla-assisted approximation over the basis of metaplectic gates based on cost-optimizing determinations made by the synthesis algorithm.

Abstract: Methods and systems transform a given single-qubit quantum circuit expressed in a first quantum-gate basis into a quantum-circuit expressed in a second, discrete, quantum-gate basis. The discrete quantum-gate basis comprises standard, implementable quantum gates. The given single-qubit quantum circuit is expressed as a normal representation. The normal representation is generally compressed, in length, with respect to equivalent non-normalized representations. The method and systems additionally can map normal representations to canonical-form representations, which are generally further compressed, in length, with respect to normal representations.

Abstract: Quantum circuits and circuit designs are based on factorizations of diagonal unitaries using a phase context. The cost/complexity of phase sparse/phase dense approximations is compared, and a suitable implementation is selected. For phase sparse implementations in the Clifford+T basis, required entangling circuits are defined based on a number of occurrences of a phase in the phase context in a factor of the diagonal unitary.

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: Methods for compiling single-qubit quantum gates into braid representations for non-Abelian quasiparticles described by the Fibonacci anyon model are based on a probabilistically polynomial algorithm that, given a single-qubit unitary gate and a desired target precision, outputs a braid pattern that approximates the unitary to desired precision and has a length that is asymptotically optimal (for a circuit with such property). Single-qubit unitaries that can be implemented exactly by a Fibonacci anyon braid pattern are classified, and associated braid patterns are obtained using an iterative procedure. Target unitary gates that are not exactly representable as braid patterns are first approximated to a desired precision by a unitary that is exactly representable, then a braid pattern associated with the latter is obtained.

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: The current application is directed to methods and systems which produce a design for an optimal approximation of a target single-qubit quantum operation comprising a representation of a quantum-circuit generated from a discrete, quantum-gate basis. The discrete quantum-gate basis comprises standard, implementable quantum gates. The methods and systems employ a database of canonical-form quantum circuits, an efficiently organized canonical-form quantum-circuit, and efficient searching to identify a minimum-cost design for decomposing and approximating an input target quantum operation.

Abstract: The current application is directed to methods and systems which produce a design for an optimal approximation of a target single-qubit quantum operation comprising a representation of a quantum-circuit generated from a discrete, quantum-gate basis. The discrete quantum-gate basis comprises standard, implementable quantum gates. The methods and systems employ a database of canonical-form quantum circuits, an efficiently organized canonical-form quantum-circuit, and efficient searching to identify a minimum-cost design for decomposing and approximating an input target quantum operation.

Abstract: Methods for compiling single-qubit quantum gates into braid representations for non-Abelian quasiparticles described by the Fibonacci anyon model are based on a probabilistically polynomial algorithm that, given a single-qubit unitary gate and a desired target precision, outputs a braid pattern that approximates the unitary to desired precision and has a length that is asymptotically optimal (for a circuit with such property). Single-qubit unitaries that can be implemented exactly by a Fibonacci anyon braid pattern are classified, and associated braid patterns are obtained using an iterative procedure. Target unitary gates that are not exactly representable as braid patterns are first approximated to a desired precision by a unitary that is exactly representable, then a braid pattern associated with the latter is obtained.