QML02 — Quantum Kernel Methods
| Field | Value |
|---|---|
| Domain | Quantum Machine Learning |
| System | Quantum feature map φ: x → |φ(x)⟩ ∈ ℋ; kernel K(x,x’) = |⟨φ(x)|φ(x’)⟩|² |
| Group | U(2ⁿ) acting on feature Hilbert space |
| H^k tier | H⁰ (kernel) / H¹ (feature map) / H² (advantage condition) |
| ISA | Meld (β = it) with data-dependent ORBIT structure |
| Status | Validated |
| Opcodes | ORBIT · LABEL · TWIST(x) · BIND |
| Papers | Paper 598, Paper 543, Paper 473 |
Physical system
A quantum kernel encodes classical data x ∈ ℝᵈ into a quantum state |φ(x)⟩ = U(x)|0⟩^⊗n via a data-dependent unitary U(x), then computes the inner product:
K(x, x') = |⟨φ(x)|φ(x')⟩|² = |⟨0| U(x)† U(x') |0⟩|²
This is used as the kernel in a classical support vector machine (SVM) or kernel ridge regression. The hope for quantum advantage: if U(x) encodes a feature map that is hard to compute classically, the kernel K(x,x’) gives access to a classically intractable feature space.
Expressibility (Sim et al. 2019): the range of states |φ(x)⟩ reachable by U(x) as x varies — the ORBIT of |0⟩ under the data-dependent unitary family. Entanglement capability (Sim et al. 2019): the average entanglement of |φ(x)⟩, measured by Meyer-Wallach or entanglement entropy — the H² BIND content.
The projected quantum kernel (Huang et al. 2021): instead of the full Hilbert space inner product, use the reduced density matrix on k qubits:
K_proj(x, x') = Tr[ρ_k(x) ρ_k(x')] ρ_k(x) = Tr_{n-k}[|φ(x)⟩⟨φ(x)|]
This is an ORBIT inner product on Gr(2^k, 2^n) — the Grassmannian of k-qubit reduced states inside the full n-qubit Hilbert space.
ISA reading
| QML concept | ISA translation | Tier |
|---|---|---|
| Feature map U(x)|0⟩ | ORBIT[x] on Meld ISA: data labels an ORBIT sector | H⁰ |
| Data-encoding gate U(x) | TWIST(x): data enters as TWIST eigenphase | H¹ |
| Kernel K(x,x’) = |⟨φ(x)|φ(x’)⟩|² | ORBIT inner product: transition amplitude | H⁰ |
| Projected kernel Tr[ρρ’] | ORBIT inner product on Gr(k,n) | H⁰ |
| Expressibility | ORBIT volume: fraction of Gr(2^k,2^n) reached | H⁰ |
| Entanglement capability | BIND count: H² content of |φ(x)⟩ | H² |
| Quantum advantage condition | σ₁²(x,x’) < 1−δ*: Schubert halt on data kernel matrix | H² |
| Data re-uploading | LABEL(x)∘TWIST interleaved: data as opcode argument | H⁰/H¹ |
Key insight: the quantum kernel is an ORBIT inner product on the Grassmannian. The classical kernel trick computes inner products in a reproducing kernel Hilbert space (RKHS); the quantum kernel computes inner products in a Grassmannian Hilbert space. The ISA makes this precise: the feature map is an ORBIT, the kernel is its transition amplitude, and quantum advantage is the Schubert halt condition applied to the kernel Gram matrix.
Target category
QKernel(n, 𝒳) — the category of quantum kernels on n qubits over data domain 𝒳. Objects: data points x ∈ 𝒳 with associated states |φ(x)⟩ ∈ ℋ_{2^n}. Morphisms: positive semi-definite kernel matrices K ∈ ℝ^{m×m} for m data points. The projected quantum kernel factors through the Grassmannian: 𝒳 → Gr(2^k, 2^n) → ℝ via the ORBIT map and inner product.
ISA programme
INPUT: LABEL[x ∈ 𝒳] -- classical data point
ENCODE: TWIST(x) on qubits 0..n-1 -- data-dependent rotation
BIND(entangling layer) -- H² content creation
TWIST(x) [repeated layers] -- data re-uploading
STATE: ORBIT[\|φ(x)⟩ = U(x)\|0⟩] -- quantum feature state
KERNEL: LABEL[K(x,x') = \|⟨φ(x)\|φ(x')⟩\|²] -- ORBIT inner product
PROJECT: ORBIT[ρ_k(x) ∈ Gr(2^k, 2^n)] -- projected kernel
CLASSIFY: LABEL[f(x) = Σᵢ αᵢ K(x,xᵢ)] -- SVM decision function
HALT: SNAP[σ₁²(K) < 1−δ*] -- quantum advantage check
β-plane position and the thermal kernel
The standard quantum kernel lives at β = it (Meld ISA, pure quantum). But the ISA reveals a richer structure: the kernel has a natural thermal extension.
Define the thermal quantum kernel at inverse temperature β:
K_β(x, x') = Tr[e^{-β H(x)} · e^{-β H(x')}] / (Z(x) Z(x'))
where H(x) is the data-dependent Hamiltonian and Z(x) = Tr[e^{-βH(x)}] is the partition function. At β = it this reduces to the standard quantum kernel. At β real this is a classical thermal kernel — computable by MCMC and related to the Gram matrix of the Gibbs distribution.
β-plane positions of the kernel family:
Im(β)
│ ← Standard quantum kernel K_β at β = it
│
──┼─────────────────────────── Re(β)
│ ← Thermal kernel K_β at β real
│ β→0: uniform kernel (useless)
│ β=β*: Forge snap — maximum information kernel
│ β→∞: classical hard kernel (argmax feature map)
The optimal kernel lives near β* on the real axis: this is the Forge ISA snap point where the partition function Z(x) has maximum curvature — the kernel is most discriminative per unit circuit cost. This is the QML analogue of the 0.234 optimal acceptance rate in MCMC (ST01).
The unified classical-quantum ML model
The ISA provides a single model that contains classical ML and quantum ML as special cases:
| Regime | β | Model | Training |
|---|---|---|---|
| Classical (hot) | β → 0 | Uniform / max entropy | No learning |
| Classical (Forge) | β real, finite | Energy-based model (EBM) | MCMC / contrastive divergence |
| Classical (Origami) | β → ∞ | Hard decision tree / SVM | Gradient descent |
| Transition | β = β* | Snap event — phase transition | Free energy minimisation |
| Quantum (Meld) | β = it | Quantum kernel / PQC | Parameter shift rule |
| Complex (Forge+Meld) | β = σ + it | Dissipative QML | β-annealing |
| p-adic | β ∈ ℚ_p | Ultrametric kernel | Hensel lifting |
The unified training trajectory is a path in the β-plane:
1. START: β real, large (Origami — structured initialisation)
2. WARM: β → β* (Forge snap — commit to right ORBIT sector)
3. WICK: β = β* → iβ* (Wick rotation — engage quantum coherence)
4. OPTIMISE: parameter shift on Meld ISA (gradient now O(1))
5. HALT: SNAP when σ₁²(K) < 1−δ* (quantum advantage confirmed)
This eliminates barren plateaus (step 2 commits before step 3 engages), provides a principled initialisation, and gives a structural criterion for when quantum advantage is genuine (step 5).
Expressibility and ORBIT volume
Sim et al. (2019) define expressibility as the deviation of the PQC’s state distribution from the Haar measure:
Expr = D_KL(P_{PQC}(F) || P_Haar(F))
| where F = | ⟨φ | |ψ⟩ | ² is the fidelity between random circuit outputs. |
ISA reading: expressibility is ORBIT volume — the fraction of the Grassmannian Gr(1, 2^n) ≅ ℂP^{2^n−1} reached by the feature map. High expressibility = large ORBIT volume = PQC approaches Haar measure. But high expressibility also means barren plateaus (QML01): the ORBIT is too large to navigate by gradient descent.
The ISA resolves this tension: high expressibility is valuable for the Forge ISA pre-training phase (large ORBIT = many sectors to explore), but the circuit should then snap to a low-dimensional ORBIT sector (low expressibility, targeted, gradient-navigable) before Wick-rotating to the Meld ISA.
Quantum advantage: the Schubert halt theorem applied to kernels
From Paper 598: the OPU halts (achieves quantum advantage) when:
σ₁²(Gram matrix) < 1 − δ*
Applied to quantum kernels: compute the SVD of the kernel Gram matrix K ∈ ℝ^{m×m}. If the leading singular value satisfies σ₁² ≥ 1−δ, the kernel is in the H⁰ ORBIT sector — it can be approximated by a classical kernel (RBF, polynomial) without quantum overhead. If σ₁² < 1−δ, the kernel has genuine H² BIND content and a classical approximation requires exponential resources.
This gives a pre-screening criterion for quantum kernel methods: compute the SVD of K before training. If the Schubert halt condition fails, switch to a classical kernel — the quantum circuit is unnecessary. If it holds, quantum advantage is structurally guaranteed.
Validation
- Schuld & Killoran (2019): quantum feature spaces and parameter-shift rule. The H¹ TWIST eigenphase gradient is exact.
- Havlíček et al. (2019): quantum kernel SVM with classically intractable feature spaces. First experimental demonstration of a quantum kernel on IBM hardware.
- Huang et al. (2021): projected quantum kernels — classical shadows give efficient approximations when the reduced state ρ_k is in the H⁰ ORBIT sector. The ISA: classical shadows succeed when σ₁²(ρ_k) ≥ 1−δ*.
- Sim et al. (2019): expressibility and entanglement capability. These are ORBIT volume and BIND count in ISA language.
- Thanasilp et al. (2022): exponential concentration of quantum kernels — the kernel values concentrate around 1/2^n exponentially in n, making them useless for classification. ISA reading: the kernel has degenerated to the Ambient (β → 0), maximum entropy ORBIT.
Connection to ZX calculus
ZX calculus (Coecke & Duncan 2008) is the complete diagrammatic rewriting system for the stabiliser fragment of the Meld ISA — the β = it, Clifford (fourth-roots-of-unity) slice. It can represent and simplify any quantum feature map built from Clifford gates.
However, ZX has no notion of:
- Temperature β: two ZX-equal circuits are Gibbs-inequivalent at finite β
- The kernel’s thermal extension: K_β at real β is invisible to ZX
- The Schubert halt condition: quantum advantage requires H² BIND content, which ZX cannot detect (ZX lives in the stabiliser H⁰ sector)
The ISA is the thermodynamic extension of ZX that captures all three.
Part of the ISA Zoo. See also: QML01 — Barren Plateaus · ML01 — Transformer Attention · β is a coordinate