Decoding Quantum Circuits: A Unified Framework for Efficient Simulation on Classical Computers

This article presents a unified framework for understanding the structure of quantum circuits, specifically Clifford and matchgate circuits, that can be efficiently simulated on a classical computer. The approach involves analyzing the operator spread within a network of basis operators during the evolution of a quantum circuit. The complexity of a calculation is quantified by the number of operators with amplitude above a threshold value. The study provides a new perspective on the simulation of quantum circuits and can be adapted into a numerical procedure where errors can be consistently controlled as a function of the simulation’s complexity.

Introduction to Quantum Circuits and Their Simulation

Quantum circuits, specifically those consisting of Clifford and matchgates, are known to be efficiently simulatable on a classical computer. This article introduces a unified framework that transparently shows the special structure that allows these circuits to be efficiently simulated. The approach relies on analyzing the operator spread within a network of basis operators during the evolution of a quantum circuit. The complexity of a calculation is quantified by the number of operators with amplitude above a threshold value. The complexity curve typically involves an initial exponential growth, saturation, and then exponential decay in the presence of decoherence. This approach can be adapted into a numerical procedure where errors can be consistently controlled as a function of the simulation’s complexity.

Quantum Circuits and Classical Computers

Calculating the time dynamics of a quantum circuit on a classical computer is generally a difficult task. This difficulty is closely related to the challenge of calculating time dynamics of generic quantum many-body problems. This difficulty is exploited in the context of quantum computational advantage, where the task is to simulate time dynamics of a quantum system. In the seminal result by the Google collaboration, a quantum random circuit was used to show that equivalent results would be difficult to obtain using classical simulation.

Exceptions in Quantum Circuits

For qubit-based quantum circuits, there are two notable exceptions. The first is the foundational result of Gottesman and Knill, which states that any quantum circuit that consists only of Clifford gates can be efficiently computed classically. The second consists of circuits involving matchgates, which are defined as any unitary operation of a specific form. For a circuit consisting purely of matchgates on nearest neighbor qubits in a one-dimensional geometry and starting from an input state that is a product state, the expectation values in the computational basis can be calculated efficiently.

Understanding the Structure of Efficiently Simulatable Circuits

A natural question that arises is whether there is any connection between such efficiently simulatable circuits. Better understanding the structure of such efficiently simulatable circuits may provide additional tools to discover other such circuits. Even for circuits which are not efficiently simulatable in the general case, in certain circumstances the complexity may only grow slowly, making them tractable in certain regimes. Another way that a circuit may become effectively tractable is by the presence of decoherence and other imperfections.

Operator Growth in Quantum Circuits

In this paper, a generic sequence of unitary operations in a quantum circuit in the presence of decoherence that scales with the depth of the circuit is examined. The complexity of simulating quantum dynamics is studied in the context of operator growth. During the evolution, an initially local operator transforms into a superposition of numerous nonlocal operators in the Heisenberg picture, thereby increasing the computational cost. Recent studies have explored universal patterns of operator growth, linking it with out-of-time-order correlators and information scrambling in various systems.

Conclusion

In conclusion, this paper presents a unified framework to understand both Clifford, matchgate, and other circuits which become effectively tractable. The study of the complexity of simulating quantum dynamics in the context of operator growth provides a new perspective on the simulation of quantum circuits. This approach can be adapted into a numerical procedure where errors can be consistently controlled as a function of the simulation’s complexity.