We construct a constant-degree and constant-gap quantum expander on $n$ qubits where each unitary can be implemented by a depth-$1$ and 1D circuit of Pauli or CNOT gates. We provide two applications of this expander. First, we use it to construct a family of frustration-free 1D Hamiltonians whose ground states obey the entanglement-gap relation $S = \Theta(\Delta^{-1/2})$; this is believed to be optimal, but achieving it had been open. Second, we use it to provide a streaming protocol that tests for closeness to a class of 1D volume-law entangled states. Moreover, we extend our quantum expander to a constant-degree and constant-gap expander on the unitary group where each unitary is a single $T$ gate, a single $T^{\dagger}$ gate, or a depth-$1$ Clifford circuit. This implies that a random sequence of unitaries from the expander yields a gapped walk on a dense subgroup of the unitary group. This improves upon previous work by Bourgain and Gamburd which did not control the dependence of the gap on the dimension.
It is well-known that every $n$-qubit unitary can be implemented by a $2^{O(n)}$-depth quantum circuit using single- and two-qubit gates. It has been open whether exponential depth is *necessary* for general unitaries, even when allowing an unlimited number of ancilla qubits. Here we show, perhaps surprisingly, that al...
Barak Nehoran, Joseph Slote, Henry S. Yuen· 1 citation· ⚡1
In the conventional theory of quantum computation, universality is discussed in terms of properties of unitary gate sets. In many experimental setups, however, we instead have access to a parametrized set of Hamiltonians, which can be exponentiated for any desired time. Accordingly, we formulate a continuous analogue o...
Drawing on insights from stochastic calculus, geometric functional analysis, and randomized linear algebra, the approach exploits the circuit's response to variations of individual gates and requires no control over convergence to high-order unitary designs.
The results show that precise Haar-like statistics can emerge from sparse commuting dynamics with remarkably low quantum resources, with applications to randomized characterization, quantum metrology, quantum algorithms, and many-body physics.
Qing-Yue Zhang, Jun-Jie Chen, Zhou You et al.· 0 citations
The phase space instruction set is a continuous-variable universal gate set involving single-qubit rotations and qubit-dependent displacements on a single boson. Using these gates, we prove that a circuit depth $\mathrm{\Omega}(\varphi(N))$ is necessary to approximately prepare a large $N$-fold rotationally invariant S...
The output state of a 2D geometrically local shallow random quantum circuit does not have long range correlations due to its lightcone structure. But this changes if one measures a subset of the qubits: long-range entanglement can be induced by the measurement process, leading to conditional correlations between distan...
Yin-Chen Liu, Max McGinley, T. Schuster et al.· 6 citations
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