Environmental memory versus noise-resilient coding (09/28/26)

TL;DR Time-resolved broadband excitation spectroscopy—driving the qubit across a wide frequency band and reading out excitation versus frequency and delay—maps a highly structured excitation landscape with cold regions, and with postselection reconstructs the full noise power spectral density while separating quantum from classical noise. Feed-forward then reveals long-lived two-level systems (TLSs) with relaxation times from tens of microseconds to milliseconds that retain environmental memory across many gate cycles, making gate fidelity non-Markovian and protocol-dependent.

• Method: Time-resolved broadband excitation spectroscopy probes qubit excitation rather than the usual relaxation-only characterization, exposing a structured spectral landscape with cold (low-excitation) frequency windows.
• Noise reconstruction: Combined with postselection, the same spectra yield a full noise power spectral density and separate quantum from classical noise contributions.
• Long-lived TLSs: Feed-forward control exposes TLSs whose relaxation times span tens of microseconds to milliseconds, and the data link stronger qubit–TLS coupling to faster TLS relaxation.
• Hardware consequence: These intrinsic TLSs can keep exciting the qubit over many operation cycles, so the environment remembers prior dynamics and quantities such as gate fidelity become non-Markovian and protocol-dependent.
• Practical payoff: Identifying and bypassing the spectral roadblocks from long-lived TLSs becomes a concrete lever for fault-tolerant cQED hardware.

TL;DR Sheaf cohomology unifies Sipser–Spielman expander codes (classical LDPC codes from local checks on expanders) with Dijkgraaf–Witten twists (discrete gauge theories with a cocycle phase), yielding twisted sheaf gauge theories—qLDPC codes with non-Abelian $D_4$ (dihedral group of order 8) topological order. Gauging measurement of addressable logical CZ as a 0-form ${\mathbb{Z}}_2^3$ subcomplex symmetry realizes a magic-state fountain (parallel CZ-magic preparation), reaching almost-constant magic rate $\Theta (n^{1-ϵ})$ on recent sheaf-code constructions.

• Framework: Twisted sheaf gauge theory builds non-Abelian $D_4$ qLDPC codes by gauging a 0-form ${\mathbb{Z}}_2^3$ subcomplex symmetry on a sheaf complex, equivalently a topological defect network of $D_4$ patches with gapped interfaces.
• Mechanism: Addressable logical CZ gates appear as those 0-form subcomplex symmetries; gauging measurement of them implements the magic-state fountain without leaving the code family.
• Constant-rate HGP: Subdividing a 2D hypergraph-product code $[[n,\Theta (n),\Omega (n^{1/2})]]$ into a quantum sheaf code prepares $\Theta (n^{1/2})$ CZ magic states in parallel, matching the geometric code-to-manifold route of arXiv:2601.06736.
• Near-constant magic rate: With Golowich–Tamo–Zhu sheaf codes $[[n,\Theta (n^{1-ϵ}),\Omega (n^{(1-ϵ)/2})]]$ for arbitrarily small $ϵ$, the same fountain prepares $\Theta (n^{1-ϵ})$ CZ magic states in parallel.

TL;DR Oracle distillation turns many calls to a noisy unknown unitary (an oracle) into one high-fidelity call by using weak queries—deliberately soft oracle couplings that trade response strength for error correctability. For Boolean oracles this yields a noise threshold: polynomial quantum advantages, including Grover search, survive under constant-rate per-qubit depolarizing noise.

• Main result: An explicit protocol distills any Boolean oracle—an unknown phase/bit-flip unitary encoding a classical Boolean function—into a single high-fidelity oracle under either adversarial corruption of a constant fraction of input qubits or i.i.d. depolarizing noise at a constant per-qubit rate.
• Mechanism: Weak queries couple only softly to the noisy oracle, so the encoded response stays correctable; majority-style aggregation then recovers a strong effective oracle, revealing a near-optimal tension between distillation strength and correctability.
• Threshold theorem: Every Boolean oracle problem whose quantum advantage is only polynomial in the domain size—including Grover search, $k$-forrelation, and Simon’s problem—keeps a quantum advantage whenever the per-qubit depolarizing rate lies below a constant threshold.
• Grover implication: The quadratic speedup of Grover search survives nearly intact at low error rates even when almost every raw query experiences errors on one or more qubits.
• Broader scope: The construction extends to fractional and continuous-time Boolean oracles, so quantum error correction can protect unknown dynamics rather than only prescribed gates—opening a route to robust computational sensing.