Polyhedral Classical Simulators for Quantum Computation

TL;DR

The paper develops a geometric, polyhedral framework for classically simulating quantum computation by encoding state spaces as simulation polytopes and linking adaptive quantum computation models (circuit, MBQC, and QCM) through instruments. It unifies and extends established simulators (notably Gottesman–Knill) with generalized, polytope-based approaches such as stabilizer, Wigner, CNC phase-space, and universal Pauli/local Pauli samplers, providing explicit update rules and complexity considerations. By analyzing whether initial states lie inside a simulator’s polytope or can be represented as quasi-probability mixtures, the work delineates efficient estimators and universal samplers, offering a geometric roadmap for pushing the boundary of efficient classical simulation. The framework also connects to simplicial-distribution theory, highlighting foundational ties between contextuality/nonlocality and simulability, with practical simulators like CNCSim embodying the phase-space/ CNC approach.

Abstract

Quantum advantage in computation refers to the existence of computational tasks that can be performed efficiently on a quantum computer but cannot be efficiently simulated on any classical computer. Identifying the precise boundary of efficient classical simulability is a central challenge and motivates the development of new simulation paradigms. In this paper, we introduce polyhedral classical simulators, a framework for classical simulation grounded in polyhedral geometry. This framework encompasses well-known methods such as the Gottesman-Knill algorithm, while also extending naturally to more recent models of quantum computation, including those based on magic states and measurement-based quantum computation. We show how this framework unifies and extends existing simulation methods while at the same time providing a geometric roadmap for pushing the boundary of efficient classical simulation further.

Figures (4)

• Figure 1: Geometry of quantum states and simulation polytopes. In the case of an efficient estimator (shown as a diamond with blue vertices), the polytope covers only part of the quantum state space (depicted as a disk). A universal sampler (depicted as the larger polytope), by contrast, contains the entire quantum state space. Red vertices indicate iterative progress toward probabilistically simulating larger regions. The update rules specify a probabilistic choice of the next vertex---for example, remaining at the same vertex with probability $p$, or moving to another vertex with probability $1-p$.
• Figure 2: Containment relation of simulation polytopes. Dashed arrows only hold for qudits of odd local dimension.
• Figure 3: Quantum circuit. The input state is the canonical basis vector $\ket{s_1\cdots s_n}=\ket{s_1}\otimes \cdots\otimes \ket{s_n}$ with corresponding projector $\Pi^s$. The unitary transforms this state to $U\ket{s_1\cdots s_n}$, corresponding to the projector $U\Pi^s U^\dagger$. Finally, a measurement is performed with projectors $\{\Pi^r : r=r_1\cdots r_n\}$.
• Figure 4: Subgroups of the unitary group relevant to quantum computation. The first is a dense subgroup that arises in the proof of quantum universality. The second is the finite Clifford group $\operatorname{Cl}_n$, which plays a central role in the construction of stabilizer (or Clifford) circuits. The last, $G_n$, is the Pauli group—an (almost) extraspecial $2$-group underlying the stabilizer subtheory of quantum mechanics. Double lines indicate that $G_n$ is normal in $\operatorname{Cl}_n$.

Theorems & Definitions (23)

• Theorem 1
• Definition 2
• Remark 3
• Example 4
• Definition 5
• Definition 6
• Definition 7
• Example 8
• Definition 9
• Definition 10