Efficient quantum pseudorandomness with simple graph states

Abstract

Measurement based (MB) quantum computation allows for universal quantum computing by measuring individual qubits prepared in entangled multipartite states, known as graph states. Unless corrected for, the randomness of the measurements leads to the generation of ensembles of random unitaries, where each random unitary is identified with a string of possible measurement results. We show that repeating an MB scheme an efficient number of times, on a simple graph state, with measurements at fixed angles and no feed-forward corrections, produces a random unitary ensemble that is an ε-approximate t-design on n-qubits. Unlike previous constructions, the graph is regular and is also a universal resource for measurement based quantum computing, closely related to the brickwork state.

On Unitary t-Designs from Relaxed Seeds

“…Moreover, most of the applications of exact unitary t -designs can be adapted to use -approximate unitary t -designs, while retaining their efficiency [ 5 , 6 , 8 , 9 , 11 , 13 , 14 ]. Finally, efficient explicit constructions of -approximate unitary t -designs for any t are well-established both in the circuit model [ 2 , 20 ] as well the measurement-based model of quantum computing [ 11 , 21 , 22 ]. For these reasons, in this work, we will focus on -approximate t -designs.…”

“…Such relaxations have natural importance when the choice of the seed is not free for various reasons; for example, in the measurement-based approach to implementing t -designs [ 11 , 21 , 22 ] (see also [ 12 , 13 ]). There, the random selection of the unitary in the ensemble is made via a measurement—that is, relying on quantum randomness, not classical randomness.…”

Fault-tolerant quantum speedup from constant depth quantum circuits

“…We take the local stochastic quantum noise (we will also refer to this noise as local stochastic noise) model, commonly studied in the quantum error correction and faulttolerance literature [9,36,37]. Our sampling problems are built on a family of schemes essentially based on local measurements on regular graph states, which correspond to constant depth two-dimensional (2D) nearest-neighbor (NN) quantum circuits showing quantum speedup [14][15][16]18,23,38]. We show that these can be made fault tolerant in a way which maintains constant depth of the quantum circuits, albeit with large (but polynomial) overhead in the number of ancilla systems used, and at most two rounds of (efficient) classical computation during the running of the circuit.…”

“…1. Graph state |G of [38] together with the prespecified measurements in the XY plane. This graph state is composed of n rows and k columns as seen in the main text (lower part of figure), and made up of two-qubit gadgets G B (green rectangles) zoomed in at the upper part of the figure (orange circle and arrow).…”

“…The π/4 symbol is a measurement at an angle π/4 in the XY plane, similarly for π/2 and zero. In the original construction of [38], the red horizontal line is a long-range CZ; these are used periodically in |G to connect two consecutive G B gadgets acting on qubits of either the first row or the last row of |G . Here, this red horizontal line is a linear cluster of 12 qubits measured at an XY angle of zero; this is in order to make the construction nearest neighbor.…”