Quantum t-designs: t-wise independence in the quantum world

TL;DR

This work defines quantum $t$-designs as distributions over pure states whose $t$-fold tensor moments reproduce Haar moments and surveys prior constructions. It then develops efficient constructions of $\epsilon$-approximate complex projective $(t,t)$-designs for arbitrary $t$, using $t$-wise independent function families and Gaussian quadrature, with sizes $O(N^{3t})$ (and an improved $O(N^t \log^c N)$) that support efficient sampling and POVM implementation. A central result shows that an approximate $(4,4)$-design suffices to derandomize the state-distinction problem of Sen via a POVM-based measurement, with a quantitative bound $\epsilon < c f^4$ where $f=\|\rho_1-\rho_2\|_F$, thereby enabling derandomization under suitable Frobenius-distance assumptions. The paper further establishes the necessity of $(4,4)$-designs (and the insufficiency of $(2,2)$-designs) for these tasks and discusses extensions to unitary designs and other randomized protocols. Overall, the results provide efficient, derandomized alternatives to Haar randomness for quantum information processing and open several avenues for applying approximate $t$-designs to broader quantum protocols.

Abstract

A t-design for quantum states is a finite set of quantum states with the property of simulating the Haar-measure on quantum states, w.r.t. any test that uses at most t copies of a state. We give efficient constructions for approximate quantum t-designs for arbitrary t. We then show that an approximate 4-design provides a derandomization of the state-distinction problem considered by Sen (quant-ph/0512085), which is relevant to solving certain instances of the hidden subgroup problem.

Theorems & Definitions (18)

• Definition 1
• Definition 2
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Definition 3
• Theorem 5
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• Theorem 7