Speaker
Description
One potential way to achieve significant progress in quantum computing is by using quantum devices to simulate quantum systems instead of relying on classical computers to perform the simulation. However, a challenge is to demonstrate that there is a fundamental difference between simulating physical systems using classical computers versus quantum computers. Formal complexity arguments often rely on theoretical assumptions and approximations, and therefore practical implementations need to balance improvements in classical simulation and novel theoretical results with advancements in quantum hardware to maintain a meaningful gap between classical and quantum computing. Among the hardest structures to simulate classically are (ideal) random quantum circuits due to their ability to scramble initial quantum information exponentially fast through the many-body correlations using entanglement and magic (non-Clifford) resources. Other examples of dynamics believed to be hard to simulate classically constitute chaotic dynamics such as the Sachdev-Ye-Kitaev model, or BosonSampling experiments in photonic quantum computation.
A particularly elegant framework for simplifying quantum circuits known as the ZX-calculus -- a by-product of rigorously describing quantum theory from the perspective of category theory and the diagrammatic language of symmetric monoidal categories -- has recently been shown to provide outstanding results in terms of strong simulation of quantum circuits. In their work, the strategy of classical simulation rely on two basic facts: 1) when translating a generic quantum circuit (taken to be, without loss of generality, in the Clifford+$T$ representation) into the ZX-calculus and simplifying this circuit using the diagrammatic rules of the formalism one has, in the end, the appearance of cat-states, 2) those cat-states have a known stabilizer decomposition that can be substituted in the circuit. One can than iteratively use the strategy of simplifying the circuit using cat-states and their decomposition until one has to simply collect the complex numbers of the resulting final simplified graphs, with the weights provided by the stabilizer decomposition of cat states. The hardness in the simulation of quantum circuits is then characterized by the exponential number of stabilizers for a given non-Clifford cat-state. However, this technique has shown to significantly improve strong simulation upon several benchmarks.
What are cat-states? These are GHZ states for which in each qubit one apply a $T$ gate, i.e., $\vert \text{cat}_n \rangle := \frac{1}{\sqrt{2}}(\mathbb{1}^{\otimes n} + Z^{\otimes n})\vert T \rangle^{\otimes n}$. Clearly those states are non-Clifford, but they have also the structure of maximally entangled $n$-qubit states. The capability of the simulation depends on the appearance of as many such structures as possible in the simplified circuit with the largest possible connectivity and best stabilized decomposition. Note also that those cat states are not equivalent to the introduced cat-states used for error correction codes. In that case, cat states refer to a way of encoding information of a qubit state into bosonic modes, which we do not deal with in our work.
Motivated by these findings, we were interested in seeing if the structure of cat states could be suitable as well for simulation in the standard approach as opposed to the ZX-calculus. For so, we start by formally introducing an infinite family of circuit gadgets that fault-tolerantly implement quantum circuits injected by cat-states. Such a family is highly influenced by phase-gadgets of ZX-calculus. The gadget, that we refer as cat-state gadget, or simply as cat gadget, inject unitary maps having a large (exponentially growing) number of equivalent representations. We expect that the gadget itself can have a use in its own, as such gadget structures might appear as steps in many quantum information and computation protocols (e.g. in quantum metrology). We also reproduce known gadget instances, e.g., the cat gadget can be shown to implement a non-Clifford controlled-S gate.
Followed by the formal introduction of the gadgets that fault-tolerantly inject cat-states into any quantum circuit we study their properties in terms of simulation, for Clifford+$T$ circuits, other architectures such as Clifford+(non-Clifford) and the robustness of magic as a tool for estimating hardness of simulation. In particular we find numerically and theoretically two different results: first, even thought the injected unitary have exponentially many different representations, these structures are rare -- in the circuits we have considered as benchmark -- and second, in terms of classical simulation they perform significantly worse than the ZX-calculus scheme. Our findings strongly suggest that the success of the ZX-calculus classical simulation arises from the fact that the simplifications and the diagrammatic rules break the rigidity of the unitary representation allowing for cat-state structures to appear more easily, as already alluded informally by some of the recent literature in ZX-calculus.
Another perspective that cat-state gadgets might be interesting, beyond the analysis of simulation, is as a tool for introducing hardness of simulation and learned of quantum random circuits. In random circuits, two different aspects make a circuit dynamics hard to be simulated and learned -- many-body entanglement and magic. These two resources are simultaneously injected by cat-state factories. One way of estimating hardness of simulation and learning is by describing how close a circuit is from a scrambling, and possibly chaotic, dynamics. In this direction, we investigate the ability of doping random Pauli circuits (and other separable circuits) with cat-state gadget circuits in generating dynamics that is scrambling or that signal chaotic dynamics. We use a well-stablished tool for investigating those properties in quantum systems known as the out-of-time-ordered correlator (OTOC). We show that cat-state gadgets are capable, on average, of both 1) exponentially decrease the value of the OTOC, 2) avoiding the revival of the OTOC initial state, characteristic of integrable (hence non-chaotic) dynamical evolutions. These findings numerically demonstrate the ability of reaching classical hardness measured by the OTOC by doping random separable circuits with cat-state gadget circuits. However, it is evident from the structure of the averaged OTOC that such hardness is beyond the chaotic capability of fully connected ideal and universal random quantum circuits.
In summary, we describe a new class of gadgets that help understanding the differences between the standard and the ZX-calculus approach to classical simulation of quantum circuits, but beyond that, that possibly has applications of their own. In particular, we present one such application as the generation of scrambling and quasi-chaotic dynamics in random circuits that lack magic or entangling resources, such as random Pauli circuits.
