English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

It is straightforward to give a sum-over-paths expression for the transitionamplitudes of a quantum circuit as long as the gates in the circuit arebalanced, where to be balanced is to have all nonzero transition amplitudes ofequal magnitude. Here we consider the question of whether, for such circuits,the relative phases of different discrete-time paths through the configurationspace can be defined in terms of a classical action, as they are forcontinuous-time paths. We show how to do so for certain kinds of quantumcircuits, namely, Clifford circuits where the elementary systems arecontinuous-variable systems or discrete systems of odd-prime dimension. Thesetypes of circuit are distinguished by having phase-space representations thatserve to define their classical counterparts. For discrete systems, thephase-space coordinates are also discrete variables. We show that for each gatein the generating set, one can associate a symplectomorphism on the phase-spaceand to each of these one can associate a generating function, defined on twocopies of the configuration space. For discrete systems, the latter associationis achieved using tools from algebraic geometry. Finally, we show that if theaction functional for a discrete-time path through a sequence of gates isdefined using the sum of the corresponding generating functions, then it yieldsthe correct relative phases for the path-sum expression. These results arelikely to be relevant for quantizing physical theories where time isfundamentally discrete, characterizing the classical limit of discrete-timequantum dynamics, and proving complexity results for quantum circuits.

Quantum circuit dynamics via path integrals: Is there a classical action for discrete-time paths?