Compendium - every shipped proof row + preset.

hadamard-involution

pauli-involution

cnot-involution

Grover amplification

A short loop that amplifies one chosen partial above all the others. At 6 qubits it peaks audibly after 6 steps — the marked partial becomes the dominant tone.

Grover amplification

A short loop that amplifies one chosen partial above all the others. At 6 qubits it peaks audibly after 6 steps — the marked partial becomes the dominant tone.

Quantum teleportation

Move a quantum state from one qubit to another without copying it. Alice measures her two qubits, sends the two classical bits to Bob, and Bob applies one of four fixups to recover the exact state on his qubit. Discovered by Bennett, Brassard, Crépeau, Jozsa, Peres, and Wootters in 1993.

Bit-flip channel

Apply X with probability p, identity with probability 1 − p. The canonical 'random bit error' on a single qubit. Audibly, it corrupts the computational-basis encoding at rate p per qubit — the kind of error the 3-qubit repetition code was invented to fix.

3-qubit repetition code

The simplest quantum error-correcting code. Encode one logical qubit across three physical qubits (|0_L⟩ = |000⟩, |1_L⟩ = |111⟩), apply noise, then decode + correct any single bit-flip. Per Shor 1995 / Nielsen-Chuang §10.2 the corrected fidelity is 1 − 3p² + 2p³ — quadratic-in-p suppression of the linear error rate. Sound the difference between corrected and uncorrected on the bit-flip-QEC preset.

Clifford gate

Any gate that maps Pauli operators to Pauli operators by conjugation — H, S, X, Y, Z, and CNOT. The whole gate set the keyboard instrument uses to entangle held keys. Non-Clifford gates (T, controlled-rotations) need the heavier state-vector engine; the stabilizer board pauses tracking when one fires.

Born rule

How quantum states turn into sound: the loudness of each partial is the squared amplitude. Σ amp² = 1 always, so the instrument self-balances.

Grover dim7 recurrence-closure at k=15

Lov Grover (Bell Labs / NYU Tandon) showed in 1996 (*STOC* / arXiv quant-ph/9605043) that an unstructured search of N items can be amplified to probability ≈ 1 in only k* ≈ ⌊π/4·√(N/M)⌋ iterations — the canonical quadratic speedup. The *recurrence-closure* phenomenon is what happens when you keep iterating past k*: the amplitude oscillates back and the marked partials briefly dim, then re-peak at a second 'closure' k. At n=6, M=4 (the dim7 tetrad {0,3,6,9}) the second peak lands at k=15 with closed-form prob sin²(31·arcsin(1/4)) ≈ 99.96 % — the cited 'glassy' near-closure that gives the dim7 preset its restless second-wind tail.

At dim7 (n=6, four marked partials at {0, 3, 6, 9}), running the Grover loop 15 times lands a second clean peak — sin²(31·arcsin(1/4)) ≈ 99.96 % — the 'glassy' recurrence-closure that re-amplifies the diminished chord after its first peak at k=3 has decayed.

Grover whole-tone hexachord at k=2

The whole-tone scale {C, D, E, F♯, G♯, A♯} — the Debussy chord — is the cleanest rounding point in the n=6 cleanness landscape. With M=6 markers in 64 basis states, the real-valued optimal step k_real lies at distance only ≈ 0.021 from k*=2: the loop reaches peak amplification in just two iterations and the closed-form peak prob sin²(5·arcsin(√(6/64))) lands at ≈ 99.977 %. This is the n=6 cleanness landscape's small-M champion — every other M either rounds further from integer k or fails to clear 99.7 %.

Six marked partials on the whole-tone hexachord {0, 2, 4, 6, 8, 10}. With M=6 in 64 basis states the Grover loop peaks at just two steps, closed-form prob ≈ 99.977 % — the cleanest small-M rounding in the n=6 landscape.

Cleanness landscape across M ∈ {1..12}

The *cleanness landscape* (CircuitExploration.md §2) is the function P_peak(M) — the closed-form first-peak Grover probability as you sweep the marked-set cardinality M from 1 to 12 at fixed n=6. It is *not* monotone: M=6 (whole-tone hex) lands at 99.977 %, M=3 (triads) lands at 99.79 %, M=4 (dim7) lands at 96.10 % — the landscape has texture. The argmax over M ∈ {1..12} sits at M=6 by closed-form algebra, and the exhaustive sweep proves this without sampling.

At n=6 (64 partials), the first-peak Grover probability is not a smooth function of how many partials you mark. Sweeping M from 1 to 12 reveals a textured landscape with M=6 (whole-tone hex) as the closed-form argmax peak.

Grover dim7 recurrence ladder

Iterating Grover past the first peak doesn't smoothly decay — it walks a *recurrence ladder*. At dim7 (n=6, M=4) the ladder's rungs at k ∈ {3, 9, 15, 21} carry closed-form probs ≈ {96.10, 71.92, 99.96, 64.81} %. The DRIFT at k=21 is the load-bearing fingerprint: the rounding pulls the rung *down* from the previous rung at k=15 instead of monotonically up. The lattice oscillates around the marked-set fixed point and the dim7 cleanness pattern is encoded in these four rungs, not in the first peak alone.

At dim7 (n=6, M=4) the Grover loop walks a ladder of rungs at k ∈ {3, 9, 15, 21} with closed-form peaks {96.1, 71.9, 99.96, 64.8} %. The drift at k=21 (lower than k=15) is the audible signal that the recurrence lattice oscillates, not closes monotonically — the substrate behind the CircuitPurityDial.

Grover off-peak leakage spectrum

The classic Grover claim is about the *marked* amplitudes — but the 60 non-marked basis states at dim7 (n=6, M=4) carry leakage amplitude cos(7·arcsin(1/4))/√60 EVERY ONE, all 60 EXACTLY equal by the Grover symmetry. This row pins not just the total leakage mass (≈ 3.87 %) but the *shape*: the uniformity across all 60 non-marked indices. CircuitExploration.md §3.4 calls this *'grit has structure'* — the audible grit underneath the marked chord isn't random hiss, it's a uniform spectral floor whose evenness is a fingerprint of Grover's per-step rotation acting identically on the entire non-marked subspace.

Underneath the four marked partials of dim7, the 60 non-marked basis states all carry the same closed-form leakage amplitude cos(7·arcsin(1/4))/√60. Grit has structure — the residual hiss is a flat spectral floor whose uniformity is the Grover symmetry's signature.

Quantum Fourier Transform unitarity

Don Coppersmith (IBM Watson) gave the first explicit quantum circuit for the QFT in 1994 (IBM RC 19642 / arXiv quant-ph/0201067) — n single-qubit Hadamards interleaved with controlled-phase gates, scaling as O(n²) instead of the FFT's O(N·log N). Unitarity QFT†·QFT = I is the algebraic gate the Shor / HHL / amplitude-estimation family rides on. At n=3 it reduces to the cited Vandermonde Plancherel identity (Σ_k e^{2πi·jk/8} · e^{−2πi·k'k/8} = 8·δ_{j,k'}) — every off-diagonal entry of the 8×8 QFT†·QFT product sums to exactly zero by an evenly-spaced roots-of-unity cancellation, pinned at substrate ULP.

The 8×8 QFT matrix at n=3 satisfies QFT†·QFT = I exactly — the Coppersmith 1994 / NC §5.1 Vandermonde Plancherel identity, recomputed here three independent ways: closed-form Frobenius residual, explicit per-cell ket sum, and a gate-level round trip across six diverse input states.

Phase estimation deterministic recovery

Alexei Kitaev (Caltech / Landau Institute; Breakthrough Prize 2012) introduced the canonical phase-estimation algorithm in 1995 (arXiv quant-ph/9511026): for a unitary U with eigenvalue e^{2πi·φ}, n estimator qubits + an inverse QFT recover φ as an n-bit integer m = φ·2^n. NC §5.2 Eq. 5.23 gives the deterministic-recovery claim: when φ is *exactly* representable in n bits, the post-iQFT amplitude on |m⟩ is unity and every other basis state carries amplitude zero EXACT — no probabilistic threshold, no Hamming sampling. The diagonal-U test at φ ∈ {0/16, 1/16, 5/16, 11/16} pins this at four exact-n-bit anchors with n=4 estimator qubits.

When the phase φ of a diagonal unitary is exactly representable in n bits, the phase-estimation circuit recovers it with probability EXACTLY 1 — no sampling, no rounding. The 4-bit estimator pinned at φ ∈ {0/16, 1/16, 5/16, 11/16} confirms peak amplitude unity at the cited basis index with off-peak leakage at 1e-12.

Lie-Trotter / Suzuki product-formula error bound

Seth Lloyd (MIT) showed in 1996 (*Science* 273, 1073) that a Hamiltonian H = Σ_j H_j can be evolved by interleaving short evolutions of each H_j separately — the first-order Lie-Trotter formula e^{-i(A+B)·Δt} ≈ e^{-iA·Δt}·e^{-iB·Δt} with cited per-step error ≤ ½·Δt²·‖[A, B]‖_op. This is the algorithmic substrate every quantum simulator rides on. Masuo Suzuki (Tokyo) extended it to symmetric higher-order forms (1990 *Phys. Lett. A* 146, 319) with Δt³ scaling at order 2; Childs-Su 2019 (*arXiv* 1912.08854) sharpened the constants. The cited Lloyd bound at H = X + Z, Δt = 0.05 holds with 10 % headroom (residual = Δt²·‖[X,Z]‖_op = Δt² EXACT via the Pauli algebra ‖[X, Z]‖ = 2).

Trotterizing a Hamiltonian sum H = A + B into alternating short evolutions e^{-iA·Δt}·e^{-iB·Δt} carries a per-step error bounded by ½·Δt²·‖[A, B]‖_op (Lloyd 1996). For X + Z at Δt = 0.05 the bound holds with 10 % headroom, residual decays as Δt², and the Pauli commutator anchor ‖[X, Z]‖ = 2 stays EXACT — the substrate Suzuki 2nd-order rides on at Δt³ scaling.

Transverse-field Ising commutator-norm sum

The transverse-field Ising chain H = −J·Σ Z_j Z_{j+1} − h·Σ X_j is the canonical 1-D quantum-magnet model (Sachdev 2011 *Quantum Phase Transitions* §1.2). For a Trotter implementation the relevant control quantity isn't the spectrum but the *commutator sum* Σ_{j<k} ‖[T_j, T_k]‖_op — the Aaronson-Gottesman 2004 (*Phys. Rev. A* 70, 052328) symplectic counting of how many term-pairs anticommute. At n=4, J=h=1, open chain the cited sum is 12 EXACT — an integer combinatorial anchor independent of any spectral computation, the bundle-independent leg in the proof row. Suzuki's 1990 symmetric 2nd-order step is cross-pinned against an order-14 Taylor reference at Δt=0.005 with Δt³ scaling.

On the n=4 transverse-field Ising chain (J=h=1, open boundary), the symplectic commutator-norm sum Σ_{j<k} ‖[T_j, T_k]‖_op equals 12 EXACT — a pure-integer combinatorial anchor (Aaronson-Gottesman 2004) the Suzuki 2nd-order Trotter step cross-pins against an order-14 Taylor reference with Δt³ scaling.

Shor 9-qubit code corrected fidelity

Peter Shor (MIT / Bell Labs; Nevanlinna Prize 1998, Breakthrough Prize 2023) discovered in 1995 (*Phys. Rev. A* 52, R2493) the first quantum error-correcting code — the 9-qubit [[9, 1, 3]] *concatenated* code: a 3-qubit phase-flip code wrapped around a 3-qubit bit-flip code. Encoding survives any single physical-qubit Pauli error (the distance-3 property). Under independent bit-flip noise at rate p per qubit the corrected ensemble fidelity is (1 − 3p² + 2p³)³ — the cubed C.5 repetition substrate. Under depolarizing noise the leading p² coefficient is C(9, 2) = 36 EXACT (Shor 1995 / NC §10.3 Eq. 10.16). Shor's result kickstarted the entire field of fault-tolerant quantum computing.

Encode one logical qubit across nine physical qubits, apply noise, decode + correct any single bit-flip OR phase-flip. The 9-qubit Shor code's corrected fidelity is (1 − 3p² + 2p³)³ under independent bit-flip noise and the depolarizing leading coefficient is C(9, 2) = 36 EXACT — the distance-3 [[9, 1, 3]] property gives perfect recovery from all 27 single-qubit Pauli errors.

Steane 7-qubit transversal-Clifford code

Andrew Steane (Oxford) discovered in 1996 (*Proc. R. Soc. A* 452, 2551) the 7-qubit [[7, 1, 3]] *self-dual CSS* code — built directly from the classical Hamming [7, 4, 3] code via the Calderbank-Shor-Steane construction. The cited transversal-Clifford property is its load-bearing virtue: bitwise application of any Clifford gate (X, Y, Z, H, S) across all 7 physical qubits implements the *logical* Clifford on the encoded qubit (with logical S_L = S† per the Steane weight-mod-4 convention). 6 stabilizer generators per NC §10.4.3 Table 10.5, anti-commute count = 0 EXACT, |0_L⟩ stabilizer eigenvalues = +1 EXACT. The transversal-Clifford property is what makes Steane 7 the workhorse code for fault-tolerant gate-by-gate quantum computing — every Clifford gate executes without extra overhead.

Seven physical qubits encode one logical qubit. Applying H, S, X, Y, or Z bitwise across all 7 physical qubits implements the corresponding logical Clifford gate on the encoded qubit — the *transversal-Clifford* property (Steane 1996 / NC §10.4.3). 6 stabilizer generators, anti-commute count = 0, |0_L⟩ +1 eigenvalues, distance-3 single-error recovery across 21 trials.

Distance-3 rotated surface code

Alexei Kitaev (Caltech) introduced the toric code in 1997 / 2003 (*Annals of Physics* 303, 2) — a 2-D lattice of stabilizers (plaquette + vertex operators) carrying a topologically-protected logical qubit. Austin Fowler, Matteo Mariantoni, John Martinis (UCSB; Nobel Prize 2025 for superconducting-qubit experiments), and Andrew Cleland sharpened this in 2012 (*Phys. Rev. A* 86, 032324) into the *rotated surface code* — the canonical entry-point for large-scale fault-tolerant quantum hardware. The d=3 version sits on 9 data qubits with 8 stabilizer generators = 4 X-plaquettes + 4 Z-plaquettes (d² − 1 EXACT). The logical-operator distance d = 3 is pinned by exhaustive Pauli enumeration over all 262,143 non-identity Paulis on 9 data qubits — the minimum-weight non-stabilizer commutator is the cited Z_L = Z_2·Z_5·Z_8 vertical column OR X_L = X_0·X_1·X_2 horizontal row.

A 2-D toric-code descendant: 9 data qubits arranged on a rotated square lattice, 8 stabilizer generators (4 X-plaquettes + 4 Z-plaquettes), logical-operator distance d = 3. Exhaustive Pauli enumeration over 262,143 candidates confirms the minimum-weight non-stabilizer commutator IS the cited Z_L vertical column OR X_L horizontal row — the canonical scalable QEC code (Fowler 2012 / Kitaev 2003).

Cross-code logical-qubit observable

Emanuel Knill (NIST Boulder) framed in 2005 (*Nature* 434, 39) the *operator-formalism* view of QEC: every code carries logicalZ / logicalX observables that read off the encoded payload at unit fidelity for the canonical basis inputs (NC §10.5). This row is the C.11 closer / C.12 enabler — across all 3 QEC codes shipped above (Shor 9, Steane 7, surface-d3) the logical observable surface satisfies ⟨0_L|Z_L|0_L⟩ = +1 EXACT, ⟨1_L|Z_L|1_L⟩ = −1 EXACT, and ⟨+_L|X_L|+_L⟩ = +1 EXACT by the logical-basis eigenvector identities. C.12's typed JointObservable surface composes any of these with a Wavefunction-side observable to give the Circuit ⊗ Wavefunction joint readout the cross-engine Rabi audible (C.13) will route through.

Each QEC code (Shor 9, Steane 7, surface-d3) exposes logical-Z and logical-X observables that read off the encoded payload at unit fidelity: ⟨0_L|Z_L|0_L⟩ = +1, ⟨1_L|Z_L|1_L⟩ = −1, ⟨+_L|X_L|+_L⟩ = +1 — all EXACT. This is the payload-readback channel the C.12 joint observable composes with a Wavefunction-side observable to enable cross-engine Rabi readout.

joint-observable-partial-trace

Jaynes-Cummings Rabi-frequency ladder

Edwin Jaynes (Washington University) and Fred Cummings (Aerospace Corp.) framed in 1963 (*Proc. IEEE* 51, 89) the canonical fully-quantum cavity-QED model — a two-level atom coupled to a single quantized cavity mode via the rotating-wave Hamiltonian H_JC = ω·a†a + ½·ω_q·σ_z + g·(a†σ⁻ + aσ⁺). On resonance the interaction conserves total excitations, block-diagonalizing the joint Hilbert space into 2×2 manifolds {|↑, n⟩, |↓, n+1⟩} with off-diagonal coupling g·√(n+1) (the dressed-state doublets). Cummings 1965 (*Phys. Rev.* 140, A1051) sharpened the analysis: each manifold's Rabi flop oscillates at Ω_n = 2g·√(n+1) — the cited *√(n+1) ladder*, exact integer arithmetic, NOT a fitted spacing. The n=0 manifold already has Ω_0 = 2g coupling, so an EMPTY cavity still splits the qubit absorption line into a doublet — the *vacuum Rabi splitting* 2g visible *with zero photons present*. The C.13 cross-engine algebraic milestone composes Wavefunction's IMPORTED jaynesCummings.ts closed forms with the Circuit-side σ_z observable through the C.12 jointExpectationValue surface; this is the first row whose math runs through two engines and lands at substrate-ULP.

The canonical fully-quantum cavity-QED model: a two-level qubit coupled to a single quantized cavity mode through H_JC = ω·a†a + ½·ω_q·σ_z + g·(a†σ⁻ + aσ⁺). The cited Rabi-frequency ladder Ω_n = 2g·√(n+1) governs the exchange between qubit excitation and cavity Fock state |n⟩; at n=0 (empty cavity) the splitting 2g is the vacuum-Rabi doublet. C.13 reads this off the C.12 joint-observable surface to validate cross-engine algebra at substrate-ULP — the first row whose math runs through both Circuit and Wavefunction.

amplitude-damping-population-decay

thermal-relaxation-steady-state

Grover amplification

A short loop that amplifies one chosen partial above all the others. At 6 qubits it peaks audibly after 6 steps — the marked partial becomes the dominant tone.

Grover amplification

A short loop that amplifies one chosen partial above all the others. At 6 qubits it peaks audibly after 6 steps — the marked partial becomes the dominant tone.

solovay-kitaev-contraction

surface-code-d5-distance

braid-fibonacci-yang-baxter

Grover · major triad

Press a key and root + major-third + perfect-fifth voice together, in tune — the marked chord auto-focuses to its cleanest Grover peak (k* = 3) on note-on. Hold Space to step past the peak and hear the amplitude recede; scrub Grover-k to 0 for the raw quantum wash.

A major chord built from Grover amplification: root, major-third, and perfect-fifth all marked together. Each step amplifies the whole chord; at three steps it reaches its peak. Press W to rotate to IV, then V, then back home.

Grover · minor triad

Press a key and root + minor-third + perfect-fifth voice together — the minor cousin of the major-triad preset, auto-focused to its cleanest Grover peak (k* = 3) on note-on. Sustain to step past the peak.

The minor cousin of the major-triad preset. Root, minor-third, and perfect-fifth marked together; peak at three Grover steps. W rotates through i → iv → V → i.

Grover · sus4

Press a key and root + perfect-fourth + perfect-fifth voice together — the suspended chord shape, unresolved between major and minor, auto-focused to its cleanest Grover peak (k* = 3) on note-on. Sustain to step past the peak.

Suspended-fourth chord: root, perfect-fourth, perfect-fifth — neither major nor minor, the chord that sounds 'about to resolve'. Same Grover peak at three steps; W rotates through a sus-shape progression.

Grover · perfect fifth

Press a key and root + perfect fifth (1 : 3/2) voice together — the power-chord shape, auto-focused to its cleanest Grover peak (k* = 4, closed-form peak prob ≈ 99.90%) on note-on. Turn the Chord knob to rotate I → IV → V → I.

Two marked partials — root + perfect fifth (1 : 3/2) — the power-chord shape. With only two markers the Grover loop takes longer to peak (four steps) but lands at closed-form peak prob ≈ 99.90%. W rotates I → IV → V → I like the major-triad preset.

Grover · augmented triad

Press a key and a symmetric major-third stack {0, 4, 8} voices together — the augmented chord, dreamy and unresolved. Auto-focuses to k* = 3 on note-on (peak prob ≈ 99.79% — a cleaner first peak than dim7). Sustain to step past the peak.

Three marked partials on a stacked major-third {0, 4, 8} — the augmented chord, dreamy and unresolved. Closed-form peak prob ≈ 99.79% at three Grover steps — a cleaner first-peak landing than dim7 because the rounding distance is small. W rotates by half-step.

Grover · whole-tone hexachord

Press a key and six partials on the whole-tone scale {0, 2, 4, 6, 8, 10} voice together — the cleanest preset in the n=6 landscape. Auto-focuses to k* = 2 on note-on (closed-form peak prob ≈ 99.98%). The Debussy chord, in tune from the first key.

Six marked partials on the whole-tone scale {0, 2, 4, 6, 8, 10} — the Debussy chord, voiced as a Grover amplification. With six markers in 64 basis states the loop peaks at just two steps and closed-form peak prob lands at ≈ 99.98% — the cleanest preset in the n=6 cleanness landscape. W rotates between the two whole-tone scales.

Grover · dim7

Press a key and four partials at equal minor-third intervals voice together — the diminished-7th tetrad, symmetric and restless. Auto-focuses to k* = 3 on note-on (M=4 at n=6 is the landscape's dirtiest first peak, ~96% — a little residue rings under the tetrad, which suits its unresolved character). Try Grover · dim7 (n=7) for the clean first-peak version.

Diminished-seventh tetrad: four marked partials at equal minor-third intervals. Symmetric and restless, the sound of a thriller-soundtrack pivot chord. With four marked indices the Grover loop still peaks at three steps; W rotates by half-step.

Grover amplification

6 qubits, marked partial #42. Each trigger applies one Grover iteration; peak audibility at k* = 6.

6 qubits, the partial at index 42 is the secret. Each step amplifies it; by step 6 it dominates the chord. Step past 6 and it recedes again.

Grover · dim7 (n=7)

Seven qubits, four marked partials at equal minor-third intervals — the same diminished-7th tetrad the n=6 dim7 voices, but at one more qubit. The cleanness landscape flips: k_real = 4.396 lands k* = 4 with closed-form peak prob ≈ 99.90% (vs n=6's 96.10% at k* = 3). Doubled state vector (128 partials, finer partial spacing). The clean first-peak alternative to the dim7 purity dial.

The same restless diminished-seventh tetrad the n=6 dim7 voices, but voiced at one more qubit. Going from 64 partials to 128 reshuffles the cleanness math — k_real lands at 4.396 instead of 3.392, so the loop reaches its first peak at four steps with closed-form peak prob ≈ 99.90% (vs n=6 dim7's 96.10% at three steps). The clean-first-peak alternative to walking the n=6 dim7 recurrence ladder with the purity dial.

Grover · magic 24 (n=8)

Eight qubits, twenty-four marked partials on a four-octave whole-tone stack {0, 2, 4, …, 46} — the n=8 cleanness lattice's large-cardinality champion. A 24-partial chord is unworkably over-marked at n=6 (64 states) but at N=256 it lands k_real = 2.024, so k* = 2 reaches closed-form peak prob ≈ 99.978% in just two Grover steps. The largest register the instrument voices (256 partials); a glassy Debussy wash you can't build at fewer qubits. W rotates between the two whole-tone scales.

Lov Grover (Bell Labs / NYU Tandon, 1996; *STOC* / arXiv quant-ph/9605043) showed an unstructured search amplifies to near-certainty in k* ≈ ⌊π/4·√(N/M)⌋ steps, with first-peak probability sin²((2k*+1)·arcsin(√(M/N))) governed by how cleanly the real optimum rounds to the integer k*. At n=8 (N=256) the cleanness landscape gains *magic cardinalities* absent at n=6: a 24-partial marked set is the over-marked M > N/4 regime at N=64 but sits in the clean small-M/N region at N=256, landing k_real ≈ 2.024 (rounding distance ≈ 0.024) so k*=2 reaches closed-form peak prob ≈ 99.978% — the cleanest large-cardinality chord the instrument can voice.

Eight qubits, twenty-four marked partials on a four-octave whole-tone stack {0, 2, 4, …, 46} — the n=8 cleanness lattice's large-cardinality champion. Twenty-four partials would be unworkably over-marked at six qubits (64 states), but across 256 states the loop peaks at just two Grover steps with closed-form prob ≈ 99.978%. The largest register the instrument voices — a glassy Debussy wash you can't build at fewer qubits. W rotates between the two whole-tone scales.

Bell-pair chord lock

Two qubits prepared in |Φ+⟩. Partials 0 and 3 voice equally with zero phase difference — the simplest entanglement you can hear.

Two qubits prepared in the |Φ+⟩ Bell state — partials 0 and 3 play together at equal volume with no phase drift. The simplest entanglement you can hear.

GHZ-3 triplet lock

Three qubits prepared in |GHZ⟩ = (|000⟩+|111⟩)/√2. Partials 0 and 7 voice equally with zero phase difference — the 3-qubit cousin of the Bell-pair chord lock.

Three qubits in |GHZ⟩ = (|000⟩+|111⟩)/√2. Partials 0 and 7 voice equally with zero phase drift; everything in between is silent. The 3-qubit extension of the Bell pair — same chord-lock pattern, one octave wider.

QFT-3 spiral

Three qubits, eight partials at equal amplitude with phases stepping 2π/8 around the unit circle — the discrete Fourier transform's signature spiral.

Three qubits, eight partials at equal volume. The QFT imprints a phase staircase around the unit circle (0, 45°, 90°, 135°, 180°, 225°, 270°, 315°) so the partials don't all peak at the same instant — the spectrogram colours wrap around like a hue wheel.

Teleportation (Bennett 1993)

Three qubits — Alice teleports her |+⟩ payload to Bob via the canonical Bennett 1993 circuit. Two Z-measurements + two classically-conditional corrections; qubit 2 ends up in the input state at fidelity = 1 EXACT, regardless of the random measurement outcomes.

Three qubits, one famous experiment. The Hadamard on qubit 0 prepares the |+⟩ payload; the Bell-pair on qubits 1+2 sets up the entangled channel; Alice's measurements + Bob's corrections move the payload from qubit 0 to qubit 2. One Space trigger runs the whole flow. The badge shows Alice's two classical bits.

Bit-flip QEC (Shor 1995)

Three qubits, the simplest quantum error-correcting code. Encode |+⟩ across 3 physical qubits, apply bit-flip noise at p = 0.05, then decode + correct in-place — qubit 0 ends up in |+⟩ at fidelity 1 − 3p² ≈ 0.993 instead of the uncorrected 1 − p = 0.95. Each trigger samples a fresh noise pattern; the spectrogram stays locked even as the noise tries to drift it off.

Three qubits run the full QEC pipeline at one Space trigger: encode, noise, decode, correct. At p = 0.05 the corrected fidelity averages 0.993 — most single-qubit errors get caught + reversed before they corrupt the audible partial pattern. Run it next to the noise-only baseline to hear what QEC buys you.

Jaynes-Cummings cavity

First cross-engine audible preset. A 440 Hz carrier tremulous'd by the cited Eberly–Narozhny–Sanchez 1980 collapse-revival on a coherent cavity initial — a tremolo decaying to coherent quantum silence then climbing back at the cited revival time t_rev = 2π·√n̄ / g.

First cross-engine audible preset. A 440 Hz carrier tremulous'd by the cited Eberly–Narozhny–Sanchez 1980 collapse-revival on a coherent cavity initial — a tremolo decaying to coherent quantum silence (the dressed-state superposition dephases over t_c = √2/g) then climbing back from the dead at the cited revival time t_rev = 2π·√n̄/g. A sound no classical envelope generator can make — both timescales are set by the photon number, not by any LFO. The Class-2 audible follow-up to the C.13 proof row #25 algebraic milestone; the worklet drives the cited dressed-state per-sample propagator at audio rate (substrate-ULP cross-pin with bipartiteJCState).

Amplitude damping

One qubit started fully excited (|1⟩, an octave up). The cited NC §8.3.1 amplitude-damping channel — a Daley 2014 quantum-jump trajectory — relaxes it toward the ground partial |0⟩ at a γ-controlled random moment: the upper octave holds, then quantum-jumps down to the fundamental. Turn Decay (γ) up to relax sooner. A non-unital channel you can hear; rides the same closed form proof row #26 pins.

A random Pauli operator applied to a qubit with some probability — the way real-world quantum hardware decoheres. The instrument samples one Pauli per noise op (Monte Carlo single-trajectory per Daley 2014), so the average over many triggers reproduces the cited textbook superoperator.

Thermal relaxation

One qubit started excited (|1⟩, an octave up), coupled to a thermal bath. The cited Lindblad 1976 / NC §8.3.1 thermal-relaxation channel relaxes (σ⁻ at γ·(n̄+1)) AND re-excites (σ⁺ at γ·n̄), so the note telegraphs between the octave and the fundamental — a quantum-jump bubbling that settles toward the cited steady-state occupation n̄/(2n̄+1). Decay (γ) sets the rate, Bath (n̄) sets how much it keeps climbing back. Rides the same closed form proof row #27 pins.

A random Pauli operator applied to a qubit with some probability — the way real-world quantum hardware decoheres. The instrument samples one Pauli per noise op (Monte Carlo single-trajectory per Daley 2014), so the average over many triggers reproduces the cited textbook superoperator.

Solovay-Kitaev

One qubit, two partials. The Depth knob compiles a fixed cited target rotation into a Clifford+T word (Dawson-Nielsen 2006) and ships it to the voice — as depth↑ the upper partial settles toward a 2:1 power split with the fundamental. You hear the gate-compilation recursion converge. Pairs with the /circuit proof page's 3/2-contraction row.

A reversible operation that rotates the qubit blend. Different gates twist the partials in different ways; combine them to compose a melody.

TFIM drift

Four qubits, sixteen partials. A held note evolves continuously under the cited transverse-field Ising Hamiltonian (Sachdev 2011 §1.2) — the Trotterized time-evolution e^{-iH·t} sloshes population across the partial bank as the transverse field flips bits and the ZZ-bonds phase them. The Drift knob scrubs how fast simulated time advances; turn it to 0 to freeze the slosh. Pairs with the /circuit proof page's Lie-Trotter + TFIM-commutator rows.

The transverse-field Ising chain H = −J·Σ Z_j Z_{j+1} − h·Σ X_j is the canonical 1-D quantum-magnet model (Sachdev 2011 *Quantum Phase Transitions* §1.2). For a Trotter implementation the relevant control quantity isn't the spectrum but the *commutator sum* Σ_{j<k} ‖[T_j, T_k]‖_op — the Aaronson-Gottesman 2004 (*Phys. Rev. A* 70, 052328) symplectic counting of how many term-pairs anticommute. At n=4, J=h=1, open chain the cited sum is 12 EXACT — an integer combinatorial anchor independent of any spectral computation, the bundle-independent leg in the proof row. Suzuki's 1990 symmetric 2nd-order step is cross-pinned against an order-14 Taylor reference at Δt=0.005 with Δt³ scaling.

On the n=4 transverse-field Ising chain (J=h=1, open boundary), the symplectic commutator-norm sum Σ_{j<k} ‖[T_j, T_k]‖_op equals 12 EXACT — a pure-integer combinatorial anchor (Aaronson-Gottesman 2004) the Suzuki 2nd-order Trotter step cross-pins against an order-14 Taylor reference with Δt³ scaling.

Logical readout (surface-d3)

One qubit, two partials — but they voice the LOGICAL qubit of the cited distance-3 rotated surface code (Fowler 2012), not its 9 physical qubits. The Logical knob morphs the encoded logical state; the Syndrome knob injects a single-qubit error; the Correct toggle runs the decoder. With correction on, the readout holds steady no matter which error strikes (distance-3 recovery); turn it off and a support error flips the two partials. Hear what quantum error correction buys you. Pairs with the /circuit proof page's surface-code + cross-code logical-observable rows.

One qubit, two partials — but they voice the LOGICAL qubit of the distance-3 rotated surface code (Fowler 2012), reduced from its 9 physical qubits. The whole encode → error → decode pipeline runs offline; only the 1-qubit logical-Z reading is shipped to the voice, so the two partials' loudness split tracks ⟨Z_L⟩. Turn Logical (θ) to morph the encoded state; turn Syndrome to inject a single-qubit error; toggle Correct to run the decoder. With Correct on, the readout never budges no matter which error strikes — that's distance-3 error correction. Turn Correct off and an error on the logical-Z support (qubits 2, 5, 8) flips the two partials. Hear what quantum error correction buys you — pairs with the /circuit proof page's surface-code + cross-code logical-observable rows.

Braid weave (Fibonacci)

One logical qubit woven by a cited 3-anyon Fibonacci braid (Nayak 2008 / Freedman-Kitaev-Larsen-Wang 2003). A B₃ braid word's per-letter R-matrices act on the qubit one crossing at a time — the two partials' balance morphs as the weave plays out (partial loudness IS the Born-rule population). Because Fibonacci braiding is universal (dense in SU(2)), a non-central weave never exactly repeats. The Weave knob picks the braid word, Rate scrubs how fast it plays. The audible companion to the /circuit proof page's Fibonacci-anyon Yang-Baxter row.

One logical qubit voiced by a topological braid. Three Fibonacci anyons fuse two ways, so their fusion space is one qubit; braiding them around each other is a quantum gate (Nayak 2008 / Freedman-Kitaev-Larsen-Wang 2003). A cited 3-strand braid word's per-crossing R-matrices act on the qubit one letter at a time, and the two partials' balance morphs as the weave plays out — partial loudness IS the Born-rule population |⟨k|ψ⟩|², so you literally hear the topological gate act. Because Fibonacci braiding is universal (dense in SU(2)), a non-central weave never exactly repeats — a bounded but endlessly-evolving timbre. The Weave knob picks the cited braid word (the Garside full twist Δ² that generates the centre of B₃, or denser alternating excursions); Rate scrubs how fast it plays. The Class-2 audible companion to the /circuit proof page's Fibonacci-anyon Yang-Baxter row (#32).