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arXiv:2305.19988 · New Journal of Physics

Non-stabilizerness and entanglement from cat-state injection

Filipa C. R. Peres, Rafael Wagner, Ernesto F. Galvão

Published Jan 1, 2024• 2 citations
arXivHTMLPDF

Abstract

Abstract Recently, cat states have been used to heuristically improve the runtime of a classical simulator of quantum circuits based on the diagrammatic ZX-calculus. Here we investigate the use of cat-state injection within the quantum circuit model. We explore a family of cat states, c a t m ∗ , and describe circuit gadgets using them to concurrently inject non-stabilizerness (also known as magic) and entanglement into any quantum circuit. We provide numerical evidence that cat-state injection does not lead to speed-up in classical simulation. On the other hand, we show that our gadgets can be used to widen the scope of compelling applications of cat states. Specifically, we show how to leverage them to achieve savings in the number of injected qubits, and also to induce scrambling dynamics in otherwise non-entangling Clifford circuits in a controlled manner.

DOI
10.1088/1367-2630/ad1b80

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Abstract Recently, cat states have been used to heuristically improve the runtime of a classical simulator of quantum circuits based on the diagrammatic ZX-calculus. Here we investigate the use of cat-state injection within the quantum circuit model. We explore a family of cat states, c a t m ∗ , and describe circuit gadgets using them to concurrently inject non-stabilizerness (also known as magic) and entanglement into any quantum circuit. We provide numerical evidence that cat-state injection does not lead to speed-up in classical simulation. On the other hand, we show that our gadgets can be used to widen the scope of compelling applications of cat states. Specifically, we show how to leverage them to achieve savings in the number of injected qubits, and also to induce scrambling dynamics in otherwise non-entangling Clifford circuits in a controlled manner.
Entanglement and non-stabilizerness are complementary resources underlying the complexity of quantum states, yet extracting either quantity from large quantum circuits generally requires exponentially scaling resources. In diagrammatic approaches to quantum computation, such as the ZX-calculus, both quantities likewise typically require expensive tensor contractions that are not native to graphical rewriting. Here we show that both entanglement and magic can be efficiently estimated directly from ZX-diagrams. Flow enables efficient extraction of a normal form separating a potentially extensively entangled graph-state backbone from non-Clifford Pauli gadgets. This structure yields additive upper and lower bounds on bipartite entanglement entropy, which we further tighten through a preprocessing procedure that removes or merges redundant non-Clifford contributions. The same normal form gives an upper bound on logarithmic stabilizer extent. We benchmark these methods in random unitary and monitored circuits, and a circuit combining Trotterized Hamiltonian evolution with Clifford layers, finding that the resulting bounds remain informative even at large qubit numbers, circuit depths, and internal spider counts. Our results provide scalable probes of entanglement and non-stabilizerness of ZX-diagrams and establish a bridge between ZX-calculus and quantum many-body resource characterization, with applications across quantum computing, quantum information, and condensed-matter physics.
Abstract We introduce an enhanced technique for strong classical simulation of quantum circuits which combines the ‘sum-of-stabilisers’ method with an automated simplification strategy based on the ZX-calculus. Recently it was shown that quantum circuits can be classically simulated by expressing the non-stabiliser gates in a circuit as magic state injections and decomposing them in chunks of 2–6 states at a time, obtaining sums of (efficiently-simulable) stabiliser states with many fewer terms than the naive approach. We adapt these techniques from the original setting of Clifford circuits with magic state injection to generic ZX-diagrams and show that, by interleaving this ‘chunked’ decomposition with a ZX-calculus-based simplification strategy, we can obtain stabiliser decompositions that are many orders of magnitude smaller than existing approaches. We illustrate this technique to perform exact norm calculations (and hence strong simulation) on the outputs of random 50- and 100-qubit Clifford + T circuits with up to 70 T-gates as well as a family of hidden shift circuits previously considered by Bravyi and Gosset with over 1000 T-gates.
Abstract Magic, also known as nonstabilizerness, quantifies the distance of a quantum state to the set of stabilizer states, and it serves as a necessary resource for potential quantum advantage over classical computing. In this work, we study magic in a measurement-only quantum circuit with competing types of Clifford and non-Clifford measurements, where magic is injected through the non-Clifford measurements. This circuit can be mapped to a classical model that can be simulated efficiently, and the magic can be characterized using any magic measure that is additive to the tensor product of single-qubit states. Leveraging this observation, we study the magic transition in this circuit in both one- and two-dimensional lattices using large-scale numerical simulations. Our results demonstrate the presence of a magic transition between two different phases with extensive magic scaling, separated by a critical point in which the mutual magic exhibits scaling behavior analogous to entanglement. We further show that these two distinct phases can be distinguished by topological magic. In a different regime, with a vanishing rate of non-Clifford measurements, we find that the magic saturates in both phases. Our work sheds light on the behavior of magic and its linear combinations in quantum circuits, employing genuine magic measures.
We introduce the notion of dismagicker: non-Clifford unitary gate designed to reduce the non-stabilizerness (also called magic) of quantum many-body states. Although both entanglement and non-stabilizerness are fundamental quantum resources, they require distinct control strategies. While disentanglers (unitary operations that lower entanglement) are well-established in tensor network methods, analogous concept for non-stabilizerness suppression has been largely missing. In this work, we define dismagicker as non-Clifford unitary operation that actively suppresses non-stabilizerness, steering states toward classically simulatable stabilizer states. We develop optimization method for constructing dismagickers within the Matrix Product States framework. Our numerical results show that the non-stabilizerness reduction procedure, when combined with entanglement reduction steps with Clifford circuits, significantly improves the accuracy for both classical simulation of many-body systems and quantum state preparation on quantum devices. Dismagicker enriches our toolkit for the manipulation of many-body states by unifying non-stabilizerness and entanglement reduction.
It is known that a quantum circuit may be simulated with classical hardware via stabilizer state (T-)decomposition in [Formula: see text] time, given t non-Clifford gates and a decomposition efficiency [Formula: see text]. The past years have seen a number of papers presenting new decompositions of lower [Formula: see text] to reduce this runtime and enable simulation of ever larger circuits. More recently, it has been demonstrated that well-placed applications of apparently weaker (higher [Formula: see text]) decompositions can in fact result in better overall efficiency when paired with the circuit simplification strategies of ZX-calculus. In this work, we take the most generalized T-decomposition (namely vertex cutting), which achieves a poor efficiency of [Formula: see text] and identify common structures to which applying this can, after simplification via ZX-calculus rewriting, yield very strong effective efficiencies [Formula: see text]. By taking into account this broader scope of the ZX-diagram and incorporating the simplification facilitated by the well-motivated cuts, we derive a handful of efficient T-decompositions whose applicabilities are relatively frequent. In benchmarking these new ‘dynamic’ decompositions against the existing alternatives, we observe a significant reduction in overall [Formula: see text] and hence overall runtime for classical simulation, particularly for certain common circuit classes.
This Letter augments the recently introduced stabilizer tensor network (STN) protocol with magic state injection, reporting a new framework with significantly enhanced ability to simulate circuits with an extensive number of non-Clifford operations. Specifically, for random T-doped N-qubit Clifford circuits the computational cost of circuits prepared with magic state injection scales as O[poly(N)] when the circuit has t≲N T gates compared to an exponential scaling for the STN approach, which is demonstrated in systems of up to 200 qubits. In the case of the hidden bit shift circuit, a paradigmatic benchmarking system for extended stabilizer methods with a tunable amount of magic, we report that our magic state injected STN framework can efficiently simulate 4000 qubits and 320T gates. These findings provide a promising outlook for the use of this protocol in the classical modeling of quantum circuits that are conventionally difficult to simulate efficiently.