Efficient Unitary Designs with a System-Size Independent Number of Non-Clifford Gates

Area of Science:

• Quantum Information Science
• Quantum Computing
• Quantum Cryptography

Background:

• Quantum information protocols often require random unitaries.
• Producing true Haar-random unitaries is computationally expensive.
• Unitary t-designs approximate Haar-randomness up to the t-th moment.

Purpose of the Study:

• To determine the non-Clifford resources needed to surpass the limitations of Clifford operations in creating t-designs.
• To quantify the overhead for achieving approximate t-designs beyond Clifford circuits.

Main Methods:

• Analyzing the structure of random Clifford circuits.
• Utilizing a variant of Schur-Weyl duality for the Clifford group.
• Deriving bounds on spectral gaps of averaging operators.

Main Results:

• A polynomial number of non-Clifford gates are sufficient to create approximate t-designs.
• The required number of non-Clifford gates is independent of system size.
• Novel bounds on the convergence time of random Clifford circuits were established.

Conclusions:

• It is possible to achieve approximate t-designs with minimal non-Clifford resources.
• Random Clifford circuits with a small density of non-Clifford gates can efficiently approximate Haar-random unitaries.
• This work provides a pathway for resource-efficient implementation of random unitaries in quantum protocols.