Quantum circuits for quantum channels

Abstract

We study the implementation of quantum channels with quantum computers while minimizing the experimental cost, measured in terms of the number of Controlled-not (C-not) gates required (single-qubit gates are free). We consider three different models. In the first, the Quantum Circuit Model (QCM), we consider sequences of single-qubit and C-not gates and allow qubits to be traced out at the end of the gate sequence. In the second (RandomQCM), we also allow external classical randomness. In the third (MeasuredQCM) we also allow measurements followed by operations that are classically controlled on the outcomes. We prove lower bounds on the number of C-not gates required and give near-optimal decompositions in almost all cases. Our main result is a MeasuredQCM circuit for any channel from m qubits to n qubits that uses at most one ancilla and has a low C-not count. We give explicit examples for small numbers of qubits that provide the lowest known C-not counts.

Convex decomposition of dimension-altering quantum channels

“…It remains to investigate whether better circuit design can be achieved using other set of gates and other methods. We note that alternative circuit designs have been studied recently [30,31], which shows that classical control can further reduce the circuit costs. Comparison of circuit designs could be made while this is beyond the scope of our current work.…”

Sequence-Discriminative Training of Neural Networks

“…In construction of shallow circuits, since C-NOT gates are usually much more prone to errors than single-qubit gates, an ideal decomposition minimizes the number of C-NOTs required for a particular task. The problem of finding near optimal (in terms of the required number of C-NOT gates) circuit topologies for quantum channels from m to n qubits was considered in [41,42]. We adapt here the construction given therein to find circuit topologies for POVMs.…”

“…The procedure is as follows. Since all possible POVMs on m input qubits can be represented by a quantum channel from m to n qubits, we can find the low-cost circuit topology for POVMs, which is universal, by applying a QR decomposition and a Cosine-Sine decomposition of the quantum channel, similarly to [41,42] (Appendix A). The high level idea for proving the near optimality of the channel decomposition is then based on a parameter counting argument.…”

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].…”