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Hybrid Quantum-Classical algorithms are a promising candidate for developing uses for NISQ devices. In particular, Parametrised Quantum Circuits (PQCs) paired with classical optimizers have been used as a basis for quantum chemistry and quantum optimization problems. Training PQCs relies on methods to overcome the fact that the gradients of PQCs vanish exponentially in the size of the circuits used. Tensor network methods are being increasingly used as a classical machine learning tool, as well as a tool for studying quantum systems. We introduce a circuit pre-training method based on matrix product state machine learning methods, and demonstrate that it accelerates training of PQCs for both supervised learning, energy minimization, and combinatorial optimization., Comment: v2: Added short comparison to entanglement devised barren plateau mitigation - relevant paper missed in first submission

Tasks such as classification of data and determining the ground state of a Hamiltonian cannot be carried out through purely unitary quantum evolution. Instead, the inherent nonunitarity of the measurement process must be harnessed. Post-selection and its extensions provide a way to do this. However, they make inefficient use of time resources---a typical computation might require $O({2}^{m})$ measurements over $m$ qubits to reach a desired accuracy and cannot be done intermittently on current (superconducting-based) NISQ devices. We propose a method inspired by thermalization that harnesses insensitivity to the details of the bath. We find a greater robustness to gate noise by coupling to this bath, with a similar cost in time and more qubits compared to alternate methods for inducing nonlinearity such as fixed-point quantum search for oblivious amplitude amplification. Post-selection on $m$ ancillae qubits is replaced with tracing out $O[log\ensuremath{\epsilon}/log(1\ensuremath{-}p)]$ (where $p$ is the probability of a successful measurement) to attain the same accuracy as the post-selection circuit. We demonstrate this scheme on the quantum perceptron, quantum gearbox, and phase estimation algorithm. This method is particularly advantageous on current quantum computers involving superconducting circuits.

The surface code is a leading candidate quantum error correcting code, owing to its high threshold, and compatibility with existing experimental architectures. Bravyi et al. (2006) showed that encoding a state in the surface code using local unitary operations requires time at least linear in the lattice size $L$, however the most efficient known method for encoding an unknown state, introduced by Dennis et al. (2002), has $O(L^2)$ time complexity. Here, we present an optimal local unitary encoding circuit for the planar surface code that uses exactly $2L$ time steps to encode an unknown state in a distance $L$ planar code. We further show how an $O(L)$ complexity local unitary encoder for the toric code can be found by enforcing locality in the $O(\log L)$-depth non-local renormalisation encoder. We relate these techniques by providing an $O(L)$ local unitary circuit to convert between a toric code and a planar code, and also provide optimal encoders for the rectangular, rotated and 3D surface codes. Furthermore, we show how our encoding circuit for the planar code can be used to prepare fermionic states in the compact mapping, a recently introduced fermion to qubit mapping that has a stabiliser structure similar to that of the surface code and is particularly efficient for simulating the Fermi-Hubbard model., 17 pages, 13 figures

Tensor networks permit computational and entanglement resources to be concentrated in interesting regions of Hilbert space. Implemented on NISQ machines they allow simulation of quantum systems that are much larger than the computational machine itself. This is achieved by parallelising the quantum simulation. Here, we demonstrate this in the simplest case; an infinite, translationally invariant quantum spin chain. We provide Cirq and Qiskit code that translates infinite, translationally invariant matrix product state (iMPS) algorithms to finite-depth quantum circuit machines, allowing the representation, optimisation and evolution of arbitrary one-dimensional systems. The illustrative simulated output of these codes for achievable circuit sizes is given.