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Encoding Properties

Every quantum encoding can be characterised by a set of measurable properties that determine its suitability for a given task. The Quantum Encoding Atlas computes these properties automatically through the analysis module and exposes them via the EncodingProperties dataclass.

This page defines each property, explains what it measures, and provides intuition for how to interpret the values.

Property Summary

Property What It Measures Range Higher Is...
Expressibility Hilbert space coverage [0, +inf) More expressive
Entanglement Capability Meyer-Wallach entanglement [0, 1] More entangled
Trainability Gradient variance (barren plateaus) [0, 1] Easier to train
Circuit Depth Sequential gate layers [1, +inf) Deeper circuit
Gate Count Total quantum gates [1, +inf) More expensive
Simulability Classical simulation feasibility Boolean —

Expressibility

What it measures: How uniformly the encoding covers the space of quantum states (Hilbert space). An encoding that can only produce a small subset of possible states is less expressive than one that can reach states throughout the full space.

How it's computed: Sample random inputs, generate the corresponding quantum states, compute the distribution of pairwise fidelities, and compare against the Haar-random (maximally uniform) distribution using KL divergence.

  Low Expressibility                High Expressibility
  (e.g., Angle Encoding)           (e.g., IQP Encoding)

  States cluster in a              States spread across
  small region:                    the full Hilbert space:

       ·····                              ·   ·
      ·······                           ·   ·   ·
     ·········                        ·   ·   ·   ·
      ·······                           ·   ·   ·
       ·····                              ·   ·

Note

Lower KL divergence = higher expressibility (closer to Haar random). The analysis module reports expressibility such that lower values indicate more expressive encodings.

Entanglement Capability

What it measures: The average amount of entanglement produced across random inputs, quantified by the Meyer-Wallach measure \( Q \).

  • \( Q = 0 \): product states (no entanglement) — e.g., Angle Encoding
  • \( Q = 1 \): maximally entangled states

Why it matters: Entanglement enables quantum correlations that have no classical analogue. Encodings that produce entangled states can represent functions that separable encodings cannot.

Trainability

What it measures: Whether gradient-based optimisation is feasible. Specifically, it estimates the variance of parameter gradients — if gradients vanish exponentially with system size (a "barren plateau"), the encoding is effectively untrainable.

  High Trainability                 Low Trainability
  (gradients are informative)       (barren plateau)

  ∂L/∂θ                            ∂L/∂θ
    │   ╱╲                            │
    │  ╱  ╲   ╱╲                      │  ─────────────  ≈ 0
    │ ╱    ╲ ╱  ╲                     │
    │╱      ╲    ╲                    │
    └───────────────                  └───────────────

Rule of thumb: Shallow circuits with limited entanglement are more trainable. Deep, highly entangled circuits risk barren plateaus.

Circuit Depth and Gate Count

What they measure: The hardware cost of implementing the encoding.

  • Depth: number of sequential gate layers (determines execution time)
  • Gate count: total gates, split into single-qubit and two-qubit (CNOT) gates

Two-qubit gates are significantly noisier than single-qubit gates on current hardware, so the CNOT count is often the more relevant metric for NISQ devices.

Simulability

What it measures: Whether the encoding can be efficiently simulated by a classical computer.

  • Simulable encodings (Angle, Basis, Higher-Order Angle): A classical computer can compute measurement outcomes in polynomial time. These cannot provide quantum advantage.
  • Non-simulable encodings (IQP, Amplitude, ZZ, etc.): Classical simulation is believed to require exponential resources. These are candidates for quantum advantage.

Detection method: The atlas checks for Clifford circuits, matchgate circuits, and product-state circuits, which are known to be classically simulable.

Accessing Properties

from encoding_atlas import IQPEncoding

enc = IQPEncoding(n_features=4, reps=2)
props = enc.properties

print(props.n_qubits)        # 4
print(props.depth)            # Circuit depth
print(props.gate_count)       # Total gates
print(props.is_entangling)    # True
print(props.simulability)     # 'not_simulable'

For detailed analysis:

from encoding_atlas.analysis import (
    compute_expressibility,
    compute_entanglement_capability,
    estimate_trainability,
    count_resources,
)

See the API Reference for the full analysis interface.

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