Efficient approximate unitary t-designs from partially invertible universal sets and their application to quantum speedup

Abstract

At its core a t-design is a method for sampling from a set of unitaries in a way which mimics sampling randomly from the Haar measure on the unitary group, with applications across quantum information processing and physics.We construct new families of quantum circuits on n-qubits giving rise to ε-approximate unitary tdesigns efficiently in O(n 3 t 12 ) depth. These quantum circuits are based on a relaxation of technical requirements in previous constructions. In particular, the construction of circuits which give efficient approximate t-designs by Brandao, Harrow and Horodecki [11] required choosing gates from ensembles which contained inverses for all elements, and that the entries of the unitaries are algebraic. We reduce these requirements, to sets that contain elements without inverses in the set, and non-algebraic entries, which we dub partially invertible universal sets.We then adapt this circuit construction to the framework of measurement based quantum computation (MBQC) and give new explicit examples of n-qubit graph states with fixed assignments of measurements (graph gadgets) giving rise to unitary t-designs based on partially invertible universal sets, in a natural way.We further show that these graph gadgets demonstrate a quantum speedup, up to standard complexity theoretic conjectures. We provide numerical and analytical evidence that almost any assignment of fixed measurement angles on an n-qubit cluster state give efficient t-designs and demonstrate a quantum speedup.

Efficient Construction of Quantum Physical Unclonable Functions with Unitary t-designs

“…Further, our construction involves only nearest neighbour interaction of qubits, which makes our construction highly practical. When viewed in the measurement based quantum computation model 22,25,37 , our circuit is constant depth, which is a further indication of its potential near-term implementation.…”

“…The reason why the binary bits m i appear is because our construction is based upon an MBQC construction similar to those in 34,35,37 which have been provably shown to converge to approximate t-designs. The bits m i have a natural interpretation in this setting as the random measurement results of single qubits in some basis (see for example 25 ).…”

“…The bits m i have a natural interpretation in this setting as the random measurement results of single qubits in some basis (see for example 25 ). Although we will not go into details of MBQC constructions here, we keep this structure of the gate set to illustrate that one can go straightforwardly between both circuit and MBQC models using translations such as those in 37 . This translation may be useful in implementing our construction on different hardware, such as, for example trapped ions 38 .…”

On Unitary t-Designs from Relaxed Seeds

“…We suspect the result found here is not optimal, and can be improved. Particularly because the number of gates in the relaxed seeds introduced here grows with n and t. We conjecture that constant sized seeds such as those in [2,13] are sufficient.…”

Efficient Verification of Boson Sampling Using a Quantum Computer

“…However, the complexity required to build a universal quantum computer retaining all the advantages granted by quantum computation makes realization of such a machine impossible with current technology 5 . Hence, different sub-universal models have been realized [6][7][8][9][10] which retain the computational powers and advantages of the proposed universal model to a certain degree. These sub-universal models are assigned with specific problems, which are deemed difficult to compute using classical computer, even with certain approximation or assumptions.…”