Foundations

A code subspace with projector P exactly corrects a specified span of errors E-a precisely when P E-a† E-b P=c-ab P for every error pair.

A stabilizer code is the common fixed subspace of an abelian Pauli subgroup excluding minus identity.

Surface/toric codes measure local stabilizer checks and infer error chains from syndromes.

Fault-tolerant computation must control faults during syndrome extraction, recovery and encoded gates as well as idle storage.

Google Quantum AI's Nature study, online 9 December 2024 and in the 2025 volume, reports a distance-seven surface-code memory using 101 physical qubits with logical error 0.143%±0.003% per cycle.

BQP consists of decision problems with uniform polynomial-size quantum circuits and bounded error, using a specified finite gate model and classical input/output.

QMA gives a polynomial-time quantum verifier a polynomial-qubit witness, accepting some witness on yes-instances and rejecting every witness on no-instances with a bounded completeness/soundness gap.

Bostanci, Huang and Vaikuntanathan's 2026 manuscript constructs a classical oracle O with QMA^O not contained in QCMA^O and BQP^O/qpoly not contained in BQP^O/poly, using a code-intersection problem with strong list-recovery properties.

As reviewed on 10 October 2026, the 2026 classical-oracle separation distinguishes quantum and classical witnesses within a relativized model but leaves the ordinary QMA versus QCMA relation unresolved by that result.

A qubit pure state is a normalized vector in C² modulo global phase; a mixed state is a positive trace-one density operator.

A physical channel on a subsystem is modeled by a completely positive trace-preserving map.

A quantum circuit composes unitary gates, ancillas, channels and measurements from a specified gate set.

For an unstructured search over N alternatives with a coherent marking oracle and one marked item, amplitude amplification uses O(sqrt(N)) oracle queries for bounded success probability.

Shor's factoring algorithm reduces factoring to order finding and uses modular exponentiation, quantum phase/Fourier processing and classical continued fractions.

Von Neumann entropy is −tr rho log2 rho and equals Shannon entropy of the density operator's eigenvalues.