6-Qubit Clifford Circuits System By Bravyi, Latone, & Maslov
Creative Algorithms Expand Quantum Computation Fundamentals and Prepare for Ideal 6-Qubit Clifford Circuits
Innovative techniques that synthesise perfect 6-qubit Clifford circuits have improved quantum circuit efficiency and comprehension. Sergey Bravyi, Joseph A. Latone, and Dmitri Maslov's revolutionary paper simplifies a complex topic by providing tools and a better theoretical understanding of quantum computing.
An crucial part of quantum computation, the Clifford group studies quantum entanglement, randomised benchmarking, magic state distillation, and quantum error correction. Clifford operations can be emulated classically, but their practicality depends on circuit implementation.
Prior efforts have been limited to 4 qubits due to the Clifford group's exponential growth, making it impossible to discover the shortest, or "optimal," circuits for these operations. The search space is 13 orders of magnitude larger than prior 4-qubit synthesis efforts and roughly 4 orders of magnitude larger than Rubik's Cube. With 6 qubits, the group has a staggering 2.1 × 10^23 items.
A New Approach to an Intractable Problem
For this computational challenge, the study team created an advanced method that saves 2.1 TB of Clifford group elements in a meticulously prepared database to indirectly synthesise ideal circuits. Clifford unitarizes are classified into equivalence classes, which are groups of units with similar optimum circuit layouts. By efficiently computing a canonical representation for each class, the search space is greatly reduced. One equivalency class can represent 1.56 trillion unitarizes.
This massive database was constructed over six months on a small cluster of Intel server-class computers using pruned breadth-first search (BFS). The key optimisation goal was to reduce the CNOT gate count because two-qubit gates in superconducting circuits and trapped ions have far worse fidelity than single-qubit gates. Important choice. Thus, reducing their number improves computation fidelity. With software modifications made during the original synthesis, researchers say a full rerun can be done in two months.
Rapid Circuit Extraction
The big database extracted optimal 6-qubit Clifford circuits exceptionally well after compilation. A consumer-grade laptop can extract an arbitrary optimal 6-qubit Clifford circuit in 0.0009358 seconds, researchers found. For an enterprise-grade PC with ample RAM, this time drops to 0.0006274 seconds.
This incredible speed comes from several sophisticated “software tricks”. Eight additional bits are added to the database to sort canonical representatives by “cost” (CNOT gate count). These bits directly describe a cost-reducing generator for fast gate selection during circuit restoration. Keeping an index of every 1024th element of the larger database parts in RAM (for circuits with 9–13 gates) reduces SSD queries and saves time. These optimisations enable fast circuit construction and randomised benchmarking programs on consumer-grade hardware.
Showing Quantum Advantage and Best Designs
Besides performance, the study advanced quantum information theory. Clifford circuits have a quantum advantage over classical reversible CNOT circuits, decreasing gates from 14 to 12 instead of 8-to-7.
The database also simplified Clifford 2-design development for up to four qubits. Unitary 2-designs are important probability distributions on the unitary group that approximate the Haar (uniform) distribution and can be used as stand-ins in randomised quantum procedures including fidelity estimation, data concealment, and quantum state tomography. The researchers found optimum reduced distributions by minimising the average CNOT cost and considering Pauli mixing restrictions. It was found that the ideal Clifford 2-design for two qubits cost 1.5, whereas three and four qubits cost more.
More broadly: Quantum Gate Classification
Previous circuit optimisation attempts were substantially expanded by this work. It also aids the more ambitious objective to classify all quantum gates. In a related field, Daniel Grier and Luke Schaeffer found 57 Clifford unitarize classes. Gate sets have 'invariants' properties under circuit building operations like composition, tensor product, qubit swapping, and the use of ancillary qubits that are returned to their initial state in their classification, which expands on Clifford gates' tableau representation.
Equality (no preferred basis), degeneracy (each input impacts one output), and X-, Y-, or Z-preserving are invariants. The CNOT gate keeps X, Z, and Z, not Y. Even though Clifford operations can be imitated, quantum error correction and fault-tolerant quantum computers require them.
Bravyi, Latone, and Maslov's enormous data and handy tools help us understand and apply Clifford classes. Using Clifford circuits to improve quantum algorithms and enable fault-tolerant quantum computing requires fast synthesis of optimal circuits for these 57 classes.
