Shallow quantum circuit for generating extremely low-entangled approximate state designs

TL;DR

A new ensemble of quantum states are discovered that serve as an $\epsilon$-approximate state $t$-design while possessing extremely low entanglement, magic, and coherence, implying that for fixed $t$ and $\epsilon, entanglement, magic, and coherence do not scale with the system size.

Abstract

Random quantum states have various applications in quantum information science. We discover a new ensemble of quantum states that serve as an $ε$-approximate state $t$-design while possessing extremely low entanglement, magic, and coherence. These resources can reach their theoretical lower bounds, $Ω(\log (t/ε))$, which are also proven in this work. This implies that for fixed $t$ and $ε$, entanglement, magic, and coherence do not scale with the system size, i.e., $O(1)$ with respect to the total number of qubits $n$. Moreover, we explicitly construct an ancilla-free shallow quantum circuit for generating such states by transforming $k$-qubit approximate state designs into $n$-qubit ones without increasing the support size. The depth of such a quantum circuit, $O(t [\log t]^3 \log n \log(1/ε))$, is the most efficient among existing algorithms without ancilla qubits. A class of quantum circuits proposed in our work offers reduced cost for classical simulation of random quantum states, leading to potential applications in quantum information processing. As a concrete example, we propose classical shadow tomography using an estimator with superpositions between only two states, from which almost all quantum states can be efficiently certified by requiring only $O(1)$ measurements and classical post-processing time.

Figures (4)

• Figure 1: Overview of the generation of approximate state designs. (a) States in an ensemble $\mathcal{E}_\mathrm{sub}$ forming an approximated state $t$-design in the subspace $\mathcal{H}_\mathrm{sub}$ are mapped to states in an ensemble forming an approximate state $t$-design in the full space $\mathcal{H}$ by random injective maps $\{U_p\}$. $S_A(\ket{\phi})$ is the half-system entanglement entropy of $\ket{\phi}$. (b,c) Generating an approximate state $t$-design using (b) subsystem unitary design with random injective map, and (c) a random circuit. Generated states have (b) $O(k)$ with the subsystem size $k$ and (c) $O(d)$ entanglements with $d=\Omega(\log n)$ and the system size $n$.
• Figure 2: Circuit for generating $O(1)$-entangled state designs. '$\otimes$' in each conditional gate represents its target bit. $U_\mathrm{sub}$ represents a circuit generating state designs in a subsystem. The circuit $U_p$ for a random injective map can be implemented by (a) a random MCX gate circuit and (b) a parallel CNOT gate circuit.
• Figure : Shallow Depth Bits Randomizer
• Figure : Random Multi-Control Condition (RMCC)

Theorems & Definitions (7)

• Lemma 1: Dimension expansion of state design
• Theorem 1
• proof
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5