Analysis of the trade-off between spatial and temporal resources for measurement-based quantum computation

Abstract

In measurement-based quantum computation (MBQC), elementary quantum operations can be more parallelized than the quantum circuit model by employing a larger Hilbert space of graph states used as the resource. Thus MBQC can be regarded as a method of quantum computation where the temporal resource described by the depth of quantum operations can be reduced compared to the quantum circuit model by using the extra spatial resource described by graph states. To analyze the trade-off relationship of the spatial and temporal resources, we consider a method to obtain quantum circuit decompositions of general unitary transformations represented by MBQC on graph states with a certain underlying geometry called generalized flow. We present a method to translate any MBQC with generalized flow into quantum circuits without extra spatial resource. We also show an explicit way to unravel acausal gates that appear in the quantum circuit decomposition derived by a translation method presented in [V. Danos and E. Kashefi, Phys. Rev. A 74, 052310 (2006)] and that represent an effect of the reduction of the temporal resource in MBQC. Finally, by considering a way to deterministically simulate these acausal gates, we investigate a general framework to analyze the trade-off between the spacial and temporal resources for quantum computation.

There and back again: A circuit extraction tale

“…Our algorithm does not use ancillae. While circuit extraction algorithms for patterns containing only XYplane measurements were already known [8,24,26,45], this is the first circuit extraction algorithm for extended gflow, i.e. where patterns may contain measurements in more than one plane.…”

“…[24] an algorithm was sketched to extract an ancilla-free circuit from a XY-plane pattern with gflow. A version of this algorithm was further worked out in [45]. Independently, an algorithm for the same task was found in Ref.…”

“…Local processes can also be used to translate a pattern into a circuit. This is used, for example, to verify that the pattern represents the desired operation [15,17,20,24,45]. Conversely, a translation of a circuit into a pattern can be used to implement known algorithms in the one-way model, or it can be combined with a translation back to a circuit to trade depth against width, to parallelise Clifford operations, or to reduce the number of T gates [8,16,32].…”

Outcome determinism in measurement-based quantum computation with qudits

“…In measurement based quantum computation (MBQC), one starts with an entangled resource state (usually graph states [BR01]), and computation is carried out by sequential measurements where at each stage the measurement choice depends on previous results [RB01;RB02;DK06;DKP07]. This adaptivity is necessary to combat the randomness induced by measurements, and plays an important role both foundationally [dGK11;Rau+11] and in terms of trade-offs in optimizing computations [BK09;MHM15].…”

“…Causal flow [DK06], and its generalisation gflow [Bro+07], are graph-theoretical tools for characterising and analysing adaptivity for graph states in MBQC. They have proven a powerful tool including optimizing adaptive measurement patterns [Esl+18], translating between MBQC and the circuit picture [MHM15; Bac+21] , with application for parallelising quantum circuits [BK09], to construct schemes for the verification of blind quantum computation [FK17;Man+17], to extract bounds on the classical simulatability of MBQC [MK14], to prove depth complexity separations between the circuit and measurement-based models of computation [BK09;MHM15], to study trade-offs in adiabatic quantum computation [AMA14] and recently for applying ZX-calculus techniques [Dun+20; de +20], including to circuit compilation [Dun+20].…”