Quantum circuit design for accurate simulation of qudit channels

Abstract

We construct a classical algorithm that designs quantum circuits for algorithmic quantum simulation of arbitrary qudit channels on fault-tolerant quantum computers within a pre-specified error tolerance with respect to diamond-norm distance. The classical algorithm is constructed by decomposing a quantum channel into a convex combination of generalized extreme channels by convex optimization of a set of nonlinear coupled algebraïc equations. The resultant circuit is a randomly chosen generalized extreme channel circuit whose run-time is logarithmic with respect to the error tolerance and quadratic with respect to Hilbert space dimension, which requires only a single ancillary qudit plus classical dits.

Universal simulation of Markovian open quantum systems

“…One other possibility, already explored in detail for the single qubit case [34], would be to explicitly construct parametrised descriptions of the quantum channels appearing in the semigroup generated by an arbitrary element of the universal set. Given such an explicit parametrised family of quantum channels, the methods of [33] could be used to implement any such channel for any given time, on a minimal dilation space, through the simulation of constituent extreme channels.…”

“…Simulations on controllable quantum devices promise to be one of the most effective tools for the study of open quantum systems, and while the majority of effort over the past twenty years has focused on the development of methods for the simulation of closed quantum systems [20][21][22][23], which undergo Hamiltonian generated unitary evolution, a plethora of methods have also been developed for the quantum simulation of open quantum systems, on a wide variety of quantum devices. These methods include collision model based approaches [24][25][26][27][28], simulation algorithms designed for conventional unitary gate based universal quantum computers [29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44] and simulation algorithms designed for more general * rsweke@gmail.com quantum simulators incorporating feedback and dissipative elements in addition to unitary gates [45][46][47][48][49][50][51][52].…”

“…One natural response to this problem is via the Stinespring dilation [54]. Given any non-unitary dynamics of some particular system, it is always possible to introduce some environment, with size the square of the system size in the general case, such that the non-unitary dynamics of the system may be simulated through unitary evolution of the total system and environment [29][30][31][32][33][34]. However, it is important to note that for an arbitrary nonunitary process, there is no guarantee that the dilated unitary admits an efficient decomposition into some sequence of unitary gates from a universal set [53], and as such this strategy offers an advantage for the construction of efficient simulation algorithms only when the original non-unitary process exhibits some useful structure, such as local interactions [31].…”

“…This problem has been considered before. In particular, Wang et al have constructed a method for the simulation of arbitrary quantum channels through the simulation of extreme channels [32,33], and in effect identified such a universal set for discrete time evolution of open quantum systems. However, for systems evolving continuously in time, even in the simplest case of Markovian semigroup dynamics it is necessary to first exponentiate the generator of the semigroup in order to obtain the quantum channels describing time evolution.…”

Experimental demonstration of a measurement-based realisation of a quantum channel

“…Work on modelling and simulating such quantum systems has seen much progress recently [22][23][24], and may shed light on fundamental physical phenomena, including phase transitions in dissipative systems [25][26][27], thermalisation [28,29] and using dissipation as a resource [30,31]. In this context, the development of techniques to realise quantum channels [32,33] representing the dynamics of realistic quantum systems has seen rapid growth-most notably for single qubits [34][35][36][37][38][39][40][41] and qudits [42][43][44]. So far, however, studies have been limited to the standard quantum circuit model [45].…”