Natalie Parham

Random unitaries are a central object of study in quantum information, with applications to quantum computation, quantum many-body physics, and quantum cryptography. Recent work has constructed unitary designs and pseudorandom unitaries (PRUs) using $\Theta(\log \log n)$-depth unitary circuits with two-qubit gates. In this work, we show that unitary designs and PRUs can be efficiently constructed in several well-studied models of \emph{constant-time} quantum computation (i.e., the time complexity on the quantum computer is independent of the system size). These models are constant-depth circuits augmented with certain nonlocal operations, such as (a) many-qubit TOFFOLI gates, (b) many-qubit FANOUT gates, or (c) mid-circuit measurements with classical feedforward control. Recent advances in quantum computing hardware suggest experimental feasibility of these models in the near future. Our results demonstrate that unitary designs and PRUs can be constructed in much weaker circuit models than previously thought. Furthermore, our construction of PRUs in constant-depth with many-qubit TOFFOLI gates shows that, under cryptographic assumptions, there is no polynomial-time learning algorithm for the circuit class $\mathsf{QAC^0}$. Finally, our results suggest a new approach towards proving that PARITY is not computable in $\mathsf{QAC^0}$, a long-standing question in quantum complexity theory.

We introduce the \emph{magic hierarchy}, a quantum circuit model that alternates between arbitrary-sized Clifford circuits and constant-depth circuits with two-qubit gates ($\textsf{QNC}^0$). This model unifies existing circuit models, such as $\QACZF$ and models with adaptive intermediate measurements. Despite its generality, we are able to prove nontrivial lower bounds. We prove new lower bounds in the first level of the hierarchy, showing that certain explicit quantum states cannot be approximately prepared by circuits consisting of a Clifford circuit followed by $\textsf{QNC}^0$. These states include ground states of some topologically ordered Hamiltonians and nonstabilizer quantum codes. Our techniques exploit the rigid structure of stabilizer codes and introduce an infectiousness property: if even a single state in a high distance code can be approximately prepared by one of these circuits, then the entire subspace must lie close to a perturbed stabilizer code. We also show that proving state preparation lower bounds beyond a certain level of the hierarchy would imply \emph{classical} circuit lower bounds beyond the reach of current techniques in complexity theory. More broadly, our techniques go beyond lightcone-based methods and highlight how the magic hierarchy provides a natural framework for connecting circuit complexity, condensed matter, and Hamiltonian complexity.

Checking whether two quantum circuits are approximately equivalent is a common task in quantum computing. We consider a closely related identity check problem: given a quantum circuit U, one has to estimate the diamond-norm distance between U and the identity channel. We present a classical algorithm approximating the distance to the identity within a factor alpha = D+1 for shallow geometrically local D-dimensional circuits provided that the circuit is sufficiently close to the identity. The runtime of the algorithm scales linearly with the number of qubits for any constant circuit depth and spatial dimension. We also show that the operator-norm distance to the identity || U - I || can be efficiently approximated within a factor alpha = 5 for shallow 1D circuits and, under a certain technical condition, within a factor alpha = 2D + 3 for shallow D-dimensional circuits. A numerical implementation of the identity check algorithm is reported for 1D Trotter circuits with up to 100 qubits.