[Paper Review] Instantaneous Non-Local Computation of Low T-Depth Quantum Circuits

Speelman, Florian|arXiv (Cornell University)|Jul 1, 1998

Quantum Computing Algorithms and ArchitectureComputer Science10 references532 citations

TL;DR

This paper introduces the Heisenberg representation for quantum computation, enabling efficient classical simulation of quantum circuits composed solely of Clifford group gates and Pauli group measurements. By tracking the evolution of Pauli operators rather than full state vectors, the method reduces computational complexity from exponential to polynomial time, proving that such circuits can be perfectly simulated on classical computers—highlighting that quantum advantage arises only when non-Clifford gates are used.

ABSTRACT

Instantaneous non-local quantum computation requires multiple parties to jointly perform a quantum operation, using pre-shared entanglement and a single round of simultaneous communication. We study this task for its close connection to position-based quantum cryptography, but it also has natural applications in the context of foundations of quantum physics and in distributed computing. The best known general construction for instantaneous non-local quantum computation requires a pre-shared state which is exponentially large in the number of qubits involved in the operation, while efficient constructions are known for very specific cases only. We partially close this gap by presenting new schemes for efficient instantaneous non-local computation of several classes of quantum circuits, using the Clifford+T gate set. Our main result is a protocol which uses entanglement exponential in the T-depth of a quantum circuit, able to perform non-local computation of quantum circuits with a (poly-)logarithmic number of layers of T gates with quasi-polynomial entanglement. Our proofs combine ideas from blind and delegated quantum computation with the garden-hose model, a combinatorial model of communication complexity which was recently introduced as a tool for studying certain schemes for quantum position verification. As an application of our results, we also present an efficient attack on a recently-proposed scheme for position verification by Chakraborty and Leverrier.

Motivation & Objective

• To develop a formalism that simplifies the analysis of quantum circuits by tracking operator evolution instead of state evolution.
• To demonstrate that circuits composed only of Clifford group gates and Pauli group measurements can be efficiently simulated on classical computers.
• To clarify the boundary between classically simulable quantum computations and those requiring universal quantum advantage.
• To provide a systematic method for analyzing quantum protocols like teleportation and error correction using operator evolution.

Proposed method

• Adopt the Heisenberg picture, where operators evolve under unitary transformations via conjugation: N → UNU†.
• Use the Pauli group as a basis for tracking operator evolution, since it is closed under Clifford group operations.
• Represent each qubit’s evolution by tracking the 2n X and Z operators, each encoded in 2n+1 bits.
• Apply the Clifford group gates (Hadamard, phase, CNOT) to update the Pauli operators according to known transformation rules.
• Handle measurements by updating the stabilizer group: measure an operator A, apply correction if outcome is -1, and update all non-commuting operators via multiplication with an anticommuting stabilizer element.
• Use the stabilizer formalism to track logical operations and simulate protocols like quantum teleportation and remote CNOT.

Experimental results

Research questions

• RQ1 Can quantum circuits composed only of Clifford group gates and Pauli group measurements be efficiently simulated on classical computers?
• RQ2 How does the Heisenberg representation simplify the analysis of quantum circuits compared to state vector evolution?
• RQ3 What is the role of the stabilizer formalism in enabling efficient simulation of quantum error-correcting codes?
• RQ4 Can quantum teleportation and remote gate operations be fully analyzed and verified using operator evolution in the Heisenberg picture?
• RQ5 What is the computational boundary between classically simulable quantum circuits and those requiring universal quantum advantage?

Key findings

• Circuits composed exclusively of Clifford group gates and Pauli group measurements can be simulated in polynomial time on classical computers, as proven by Knill’s theorem.
• The Heisenberg representation reduces the complexity of analyzing n-qubit circuits from exponential to polynomial in n by tracking only 2n Pauli operators.
• The stabilizer formalism allows exact tracking of quantum states in stabilizer codes, enabling efficient simulation of quantum error correction.
• Quantum teleportation and remote CNOT operations can be fully analyzed and verified using the Heisenberg representation, with measurement outcomes determining correction operations.
• The method confirms that quantum advantage in quantum computation arises only when non-Clifford gates (e.g., T gate) are used, as Clifford-only circuits are classically simulable.