Quantum algorithm

Quantum mechanics

Diagonal

Algorithm

Quantum computer

Quantum error correction

Quantum gate

Quantum circuit

Rank (graph theory)

Superposition principle

Parallel computing

Computer science

Electronic circuit

Speedup

Physics

Mathematics

Artificial intelligence

Geometry

Quantum

Combinatorics

Qubit

Topology (electrical circuits)

Recent work has explored using the stabilizer formalism to classically simulate quantum circuits containing a few non-Clifford gates. The computational cost of such methods is directly related to the notion of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-mathit" mathvariant="italic">s</mml:mi><mml:mi class="MJX-tex-mathit" mathvariant="italic">t</mml:mi><mml:mi class="MJX-tex-mathit" mathvariant="italic">a</mml:mi><mml:mi class="MJX-tex-mathit" mathvariant="italic">b</mml:mi><mml:mi class="MJX-tex-mathit" mathvariant="italic">i</mml:mi><mml:mi class="MJX-tex-mathit" mathvariant="italic">l</mml:mi><mml:mi class="MJX-tex-mathit" mathvariant="italic">i</mml:mi><mml:mi class="MJX-tex-mathit" mathvariant="italic">z</mml:mi><mml:mi class="MJX-tex-mathit" mathvariant="italic">e</mml:mi><mml:mi class="MJX-tex-mathit" mathvariant="italic">r</mml:mi></mml:mrow></mml:math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mtext class="MJX-tex-mathit" mathvariant="italic">rank</mml:mtext></mml:mrow></mml:math>, which for a pure state<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ψ</mml:mi></mml:math>is defined to be the smallest integer<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>χ</mml:mi></mml:math>such that<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ψ</mml:mi></mml:math>is a superposition of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>χ</mml:mi></mml:math>stabilizer states. Here we develop a comprehensive mathematical theory of the stabilizer rank and the related approximate stabilizer rank. We also present a suite of classical simulation algorithms with broader applicability and significantly improved performance over the previous state-of-the-art. A new feature is the capability to simulate circuits composed of Clifford gates and arbitrary diagonal gates, extending the reach of a previous algorithm specialized to the Clifford+T gate set. We implemented the new simulation methods and used them to simulate quantum algorithms with 40-50 qubits and over 60 non-Clifford gates, without resorting to high-performance computers. We report a simulation of the Quantum Approximate Optimization Algorithm in which we process superpositions of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>χ</mml:mi><mml:mo>∼</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mn>6</mml:mn></mml:msup></mml:math>stabilizer states and sample from the full<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>-bit output distribution, improving on previous simulations which used<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>∼</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mn>3</mml:mn></mml:msup></mml:math>stabilizer states and sampled only from single-qubit marginals. We also simulated instances of the Hidden Shift algorithm with circuits including up to 64<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>T</mml:mi></mml:math>gates or 16 CCZ gates; these simulations showcase the performance gains available by optimizing the decomposition of a circuit's non-Clifford components.

Simulation of quantum circuits by low-rank stabilizer decompositions