[Paper Review] 6-qubit Optimal Clifford Circuits

Sergey Bravyi, Joseph A. Latone+1 more|arXiv (Cornell University)|Dec 11, 2020

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper presents a scalable algorithm and C++ implementation to synthesize optimal Clifford circuits for all 6-qubit Clifford group elements, achieving sub-millisecond circuit extraction via a 2.1TB database of canonical representatives. The method enables the first exhaustive synthesis of optimal 6-qubit Clifford circuits, revealing quantum advantage over CNOT-only circuits and constructing optimal 2-designs up to 4 qubits.

ABSTRACT

Clifford group lies at the core of quantum computation -- it underlies quantum error correction, its elements can be used to perform magic state distillation and they form randomized benchmarking protocols, Clifford group is used to study quantum entanglement, and more. The ability to utilize Clifford group elements in practice relies heavily on the efficiency of their circuit-level implementation. Finding short circuits is a hard problem; despite Clifford group being finite, its size grows quickly with the number of qubits $n$, limiting known optimal implementations to $n{=}4$ qubits. For $n{=}6$, the number of Clifford group elements is about $2.1{\cdot}10^{23}$. In this paper, we report a set of algorithms, along with their C/C++ implementation, that implicitly synthesize optimal circuits for all 6-qubit Clifford group elements by storing a subset of the latter in a database of size 2.1TB (1KB=1024B). We demonstrate how to extract arbitrary optimal 6-qubit Clifford circuit in $0.0009358$ and $0.0006274$ seconds using consumer- and enterprise-grade computers (hardware) respectively, while relying on this database.

Motivation & Objective

• To enable efficient, optimal synthesis of all 6-qubit Clifford group elements, which number ~2.1×10²³ and are otherwise intractable to synthesize directly.
• To overcome the exponential growth of the Clifford group by classifying unitaries into equivalence classes with similar optimal circuit structures.
• To develop a practical database and indexing system that allows real-time retrieval of optimal circuits on consumer and enterprise hardware.
• To establish new examples of quantum advantage using Clifford circuits over CNOT-only circuits and to compute optimal Clifford 2-designs for up to 4 qubits.
• To provide a scalable framework for optimal Clifford circuit synthesis that extends beyond the previously known limit of 4 qubits.

Proposed method

• The method partitions the 6-qubit Clifford group into equivalence classes based on shared optimal circuit structure, reducing the search space via symmetry and invariance under local Clifford operations.
• It employs a canonical representative for each class, computed via a systematic transformation that normalizes each unitary to a unique form for efficient database indexing.
• The algorithm uses a combination of group-theoretic techniques, symplectic matrix manipulation, and efficient data structures to precompute and store optimal circuits for all canonical representatives.
• A 2.1TB database of canonical representatives is constructed using high-performance computing (HPC), with a 2.5GB index stored in RAM for fast lookups.
• Circuit extraction is accelerated via optimized lookup: 0.0009358 seconds on consumer hardware and 0.0006274 seconds on enterprise systems.
• The approach leverages the fact that optimal circuits for equivalent classes can be reconstructed from their canonical forms using local Clifford operations.

Experimental results

Research questions

• RQ1 What is the maximum number of CNOT gates required to implement any 6-qubit Clifford unitary?
• RQ2 Can Clifford circuits provide a quantum advantage over CNOT-only circuits in terms of gate count efficiency?
• RQ3 What is the minimal average gate cost for a 2-design of 6-qubit Clifford unitaries, and how does it compare to the full Clifford group?
• RQ4 How can the 6-qubit Clifford group be partitioned into equivalence classes with isomorphic optimal circuit structures?
• RQ5 What are the optimal 2-designs for 2-, 3-, and 4-qubit Clifford groups, and what are their average gate costs?

Key findings

• The maximal CNOT gate count required to implement any 6-qubit Clifford unitary is 24, with the distribution of gate counts spanning from 0 to 24.
• The average gate cost for the full 6-qubit Clifford group is approximately 12.5, with a standard deviation of about 3.5.
• Optimal 2-designs were computed for 2, 3, and 4 qubits, with average costs of 1.5, 3.12363, and 5.08034 respectively—significantly lower than the full Clifford group's average costs.
• The method enables circuit extraction in 0.0009358 seconds on consumer-grade hardware and 0.0006274 seconds on enterprise systems, demonstrating sub-millisecond performance.
• A new example of quantum advantage is established: Clifford circuits can implement certain unitaries with fewer CNOT gates than CNOT-only circuits.
• The 2.1TB database of canonical representatives allows exhaustive synthesis of optimal 6-qubit Clifford circuits, extending the known optimal synthesis limit from 4 to 6 qubits.