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Journal of Physics A Mathematical and General

Quantum gate arrays as coherent sums over classical logic gate arrays

Bruno Nachtergaele, Vipul Periwal

Published Dec 21, 1997• 0 citations
Publisher

Abstract

Quantum spin systems may provide physical realizations of quantum gate arrays. It is shown that certain natural unitary time evolution matrices for spin- quantum spins, interpreted in this context as quantum gate arrays, can be represented as coherent sums, with appropriate phases, over classical logic gate arrays, in a direct analogy with the Feynman path integral representation of quantum mechanics. On the other hand, it is shown that a natural quantum gate obtained by analytically continuing the transfer matrix of the anisotropic nearest-neighbour Ising model to imaginary time, does not admit such a representation.

DOI
10.1088/0305-4470/30/24/027

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Quantum spin systems may provide physical realizations of quantum gate arrays. It is shown that certain natural unitary time evolution matrices for spin- quantum spins, interpreted in this context as quantum gate arrays, can be represented as coherent sums, with appropriate phases, over classical logic gate arrays, in a direct analogy with the Feynman path integral representation of quantum mechanics. On the other hand, it is shown that a natural quantum gate obtained by analytically continuing the transfer matrix of the anisotropic nearest-neighbour Ising model to imaginary time, does not admit such a representation.
It is shown that certain natural quantum logic gates, {\it i.e.} unitary time evolution matrices for spin-\frac{1}{2} quantum spins, can be represented as sums, with appropriate phases, over classical logic gates, in a direct analogy with the Feynman path integral representation of quantum mechanics. On the other hand, it is shown that a natural quantum gate obtained by analytically continuing the transfer matrix of the anisotropic nearest-neighbour Ising model to imaginary time, does not admit such a representation.
It is straightforward to compute the transition amplitudes of a quantum circuit using the sum-over-paths methodology when the gates in the circuit are balanced, where a balanced gate is one for which all non-zero transition amplitudes are of equal magnitude. Here we consider the question of whether, for such circuits, the relative phases of different discrete-time paths through the configuration space can be defined in terms of a classical action, as they are for continuous-time paths. We show how to do so for certain kinds of quantum circuits, namely, Clifford circuits where the elementary systems are continuous-variable systems or discrete systems of odd-prime dimension. These types of circuit are distinguished by having phase-space representations that serve to define their classical counterparts. For discrete systems, the phase-space coordinates are also discrete variables. We show that for each gate in the generating set, one can associate a symplectomorphism on the phase-space and to each of these one can associate a generating function, defined on two copies of the configuration space. For discrete systems, the latter association is achieved using tools from algebraic geometry. Finally, we show that if the action functional for a discrete-time path through a sequence of gates is defined using the sum of the corresponding generating functions, then it yields the correct relative phases for the path-sum expression. These results are likely to be relevant for quantizing physical theories where time is fundamentally discrete, characterizing the classical limit of discrete-time quantum dynamics, and proving complexity results for quantum circuits.
As physical systems, qubits must evolve from input to output state. We describe a simple scheme in which the effect of a quantum gate is described by the action of an effective Hamiltonian acting for some characteristic time. This model shows that the action of common unary gates is to induce Bloch sphere trajectories along lines of latitude relative to an eigenvector of the gate. Such trajectories would immediately move a `rebit', initially confined to a line of latitude, off this line and acquire a complex phase. The role of the complex phase in bringing about the entanglement of two qubits is also highlighted. It is then asked whether such dynamics could be modelled using real QM. It is shown that the continuous evolution required for such dynamics can only be provided by members of the special orthogonal group of the vector space. Since the matrices representing many quantum gates of interest have determinant -1, no real special orthogonal operators can model their evolution if the dimension of the real vector space is the same as that of the complex space. Next we look at the mapping from a complex vector space of dimension NNN to a real space of dimension 2N2N2N that is often used to construct `real' QM. It is shown that this is just an isomorphic mapping from the scalar representation of complex numbers to their 2×22\times22×2 matrix equivalents, so that the resulting matrices are actually represent complex matrices. Finally, we investigate the endomorphism of real vector spaces of dimension N=2nN = 2^{n}N=2n, where n∈Z+n \in \mathbb{Z}^{+}n∈Z+, suitable for the modelling of n−1n-1n−1 qubits. We confirm that the mapping from C2n−1→R2n\mathbb{C}^{2^{n-1}} \to \mathbb{R}^{2^{n}}C2n−1→R2n only maps elements of End(C2n−1)\mathrm{End}(\mathbb{C}^{2^{n-1}})End(C2n−1) to a restricted subspace of End(R2n)\mathrm{End}(\mathbb{R}^{2^{n}})End(R2n) that reproduces the `real' representation of complex matrices.
Abstract An interacting spin system is investigated within the scenario of the Feynman path integral representation of quantum mechanics. Short‐time propagator algorithms and a discrete time formalism are used in combination with a basis set involving Grassmann variables coherent states to get a many‐body analytic propagator. The generating function thus obtained leads, after an adequate tracing over Grassmann variables in the imaginary time domain, to the partition function. A spin 1/2 Hamiltonian involving the whole set of interactions is considered. Fermion operators satisfying the standard anticommutation relations are constructed from the raising and lowering spin operators via the Jordan–Wigner transformation. The partition function obtained is more general than the partition function of the traditional Ising model involving only first‐neighbor interactions. Computations were performed assuming that the coupling as a function of the distance can be reasonably well represented by an Airy function. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002
The Grassmann representation for the system of qubits, is considered. The treatment is based on natural description of the qubits system as fermions and uses coherent states of fermions. The quantum logic gates are represented in two forms - by symbols of operations and by partial differential operators on the symbols of the states. The considered representation of quantum logic is called as superlogic. The examples are given for classical logic operations of negation, conjunction, disjunction and for reversible three-bit Toffoli gate and its quantum generalization - three-qbit Deutsch gate. The representation for composite gates is also considered. Path integral represenatation in Grassmann algebra for quantum automaton with qubits memory is described. In particular for the autonomous automaton corresponding to the special case of general dynamic system this description differ from path integral representation in commutative memory phase space by the specific nonliner term in superaction.
Quantum computation is a continuously growing research area which is based on nature and resources of quantum mechanics, as superposition and entanglement. In its quantum circuits version, the use of convenient and appropriate gates is essential. But while those gates adopt convenient forms for computational algorithms, their design depends on specific quantum systems and stu? being used. These gates need manage quantum systems based on physical interactions ruled by quantum Hamiltonians. With this, gates design is restricted to properties and limitations of interactions and physical elements being involved. This work shows how anisotropic Ising interactions, written in a non local basis, lets reproduce elementary operations in terms of which unitary processes can be factorized. In this sense, gates could be written as a sequence of pulses ruled by that interaction driven by magnetic fields, stating alternative results in quantum gates design for magnetic systems.