Unitary $t$-designs from $relaxed$ seeds

Abstract

The capacity to randomly pick a unitary across the whole unitary group is a powerful tool across physics and quantum information. A unitary $t$-design is designed to tackle this challenge in an efficient way, yet constructions to date rely on heavy constraints. In particular, they are composed of ensembles of unitaries which, for technical reasons, must contain inverses and whose entries are algebraic. In this work, we reduce the requirements for generating an $\epsilon$-approximate unitary $t$-design. To do so, we first construct a specific $n$-qubit random quantum circuit composed of a sequence of, randomly chosen, 2-qubit gates, chosen from a set of unitaries which is approximately universal on $U(4)$, yet need not contain unitaries and their inverses, nor are in general composed of unitaries whose entries are algebraic; dubbed $relaxed$ seed. We then show that this relaxed seed, when used as a basis for our construction, gives rise to an $\epsilon$-approximate unitary $t$-design efficiently, where the depth of our random circuit scales as $poly(n,t,log(1/\epsilon ))$, thereby overcoming the two requirements which limited previous constructions. 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 ( Brandão, Harrow, and Horodecki; Commun. Math. Phys. (2016) 346: 397) are sufficient.

arXiv commentTypos corrected. Readability improved. Results unchanged. Proofs unchanged

References in corpus (27)

• Black holes as mirrors: quantum information in random subsystems
• Exact and Approximate Unitary 2-Designs: Constructions and Applications
• A fault-tolerant one-way quantum computer
• Randomizing quantum states: Constructions and applications
• Random Quantum Circuits are Approximate 2-designs
• Local random quantum circuits are approximate polynomial-designs
• Random Quantum Circuits and Pseudo-Random Operators: Theory and Applications
• Experimental demonstration of graph-state quantum secret sharing
• Fault-tolerant quantum computation with cluster states
• Approximation by Quantum Circuits
• Investigating the limits of randomized benchmarking protocols
• Approximate unitary $t$-designs by short random quantum circuits using nearest-neighbor and long-range gates
• Unitary designs and codes
• Architectures for quantum simulation showing a quantum speedup
• Thermalization and canonical typicality in translation-invariant quantum lattice systems
• Random Unitaries Give Quantum Expanders
• Resource-efficient verification of quantum computing using Serfling's bound
• Closing gaps of a quantum advantage with short-time Hamiltonian dynamics
• Measurement-based classical computation
• Efficient quantum pseudorandomness with simple graph states
• Derandomizing quantum circuits with measurement based unitary designs
• On the explicit constructions of certain unitary $t$-designs
• Random coding for sharing bosonic quantum secrets
• Testing randomness with photons
• A simple protocol for certifying graph states and applications in quantum networks
• Efficient approximate unitary t-designs from partially invertible universal sets and their application to quantum speedup
• Unitary $t$-designs from $relaxed$ seeds

Cited by in corpus (4)

• Models of quantum complexity growth
• Efficient unitary designs with a system-size independent number of non-Clifford gates
• Efficient Construction of Quantum Physical Unclonable Functions with Unitary t-designs
• Unitary $t$-designs from $relaxed$ seeds