mlx-quantum

Fast, differentiable quantum machine learning in pure Apple MLX.

Statevector simulation runs on the Metal GPU and is differentiable end-to-end through MLX autodiff — so a quantum layer trains like any other mlx.nn module, with no custom gradient code. Forward values and gradients match Qiskit to ~1e-6 (float32), and it is ~100–400× faster end-to-end than driving Qiskit's EstimatorQNN from Python. See Validation for the evidence.

Install

uv add mlx-quantum
# or
pip install mlx-quantum

Requires Python ≥ 3.13 and Apple Silicon. The library depends only on MLX and NumPy; Qiskit is optional and used solely to cross-validate/benchmark.

Quickstart

QuantumLayer is a trainable mlx.nn.Module. Drop it into a model and train:

import mlx.core as mx
import mlx.nn as nn
from mlx_quantum import QuantumLayer

class HybridMLP(nn.Module):
    def __init__(self):
        super().__init__()
        self.pre = nn.Linear(8, 4)
        self.qnn = QuantumLayer(num_qubits=4, reps=2)   # trainable quantum layer
        self.post = nn.Linear(4, 3)

    def __call__(self, x):
        x = mx.tanh(self.pre(x)) * mx.pi                # encode into rotation angles
        return self.post(self.qnn(x))

The quantum weights are ordinary MLX parameters — nn.value_and_grad and any optimizer update them automatically:

loss_and_grad = nn.value_and_grad(model, loss_fn)
loss, grads = loss_and_grad(model, x, y)
optimizer.update(model, grads)

Building custom circuits

QuantumLayer runs a hardware-efficient ansatz, but the simulator primitives are public — build any circuit as a plain differentiable function:

import mlx.core as mx
from mlx_quantum import zero_state, apply_1q, apply_2q, expval_all_z, H, ry, CX

def circuit(x, weights):            # x: (batch, n) angles, weights: (n,)
    n = x.shape[1]
    state = zero_state(x.shape[0], n)
    for q in range(n):
        state = apply_1q(state, H, q)
    for q in range(n):
        state = apply_1q(state, ry(x[:, q]), q)          # per-sample encoding
    for q in range(n):
        state = apply_1q(state, ry(weights[q]), q)        # trainable
    for q in range(n - 1):
        state = apply_2q(state, CX, q, q + 1)
    return expval_all_z(state)       # <Z> per qubit, shape (batch, n)

grads = mx.grad(lambda w: mx.sum(circuit(x, w)))(weights)  # just works

Gates provided: H, X, Y, Z, rx, ry, rz, CX, CZ. Add your own — a single-qubit gate is any (2, 2) complex mx.array; a two-qubit gate is a (2, 2, 2, 2) tensor [out0, out1, in0, in1].

How it works

A statevector is a complex mx.array of shape (batch,) + (2,) * num_qubits; qubit ordering is little-endian to match Qiskit exactly (qubit 0 is the fastest index, so flattening reproduces Qiskit's amplitude order). Gates are applied as einsum contractions, so the entire simulation is differentiable and GPU-resident. Because there is no custom vjp and no NumPy round-trip, mx.grad differentiates the circuit directly — including through complex amplitudes.

Two MLX specifics the implementation works around: the initial state is built as a constant (not an in-place assignment, which compiles to an unsupported complex GPU scatter), and gates are contractions rather than take/gather (whose backward is also a scatter).

Examples

uv run python examples/simple_mlp.py                             # hybrid MLP training
uv run --extra examples python examples/benchmark_vs_qiskit.py   # quick speed + accuracy vs Qiskit

Validation

Two tracks — is it correct, and is the speed claim fair? All measurements are noiseless statevector, float32. Regenerate with uv run --extra examples python benchmarks/validate.py (details in benchmarks/).

Correctness. Forward values and gradients are compared against Qiskit (Statevector and ReverseEstimatorGradient) over 142 random circuits covering every gate (H, X, Y, Z, rx, ry, rz, CX, CZ); per-circuit error stays at ~1e-6. (The batch-summed gradient error on the accuracy plot climbs to ~1e-5 by 8 qubits — that is float32 accumulation from summing 128 terms into one number, still ≥5 significant figures, not a modelling error.) Gates are checked for unitarity, the state norm is checked after every layer, and an asymmetric circuit pins the little-endian qubit order to Qiskit's.

Performance. Two honest baselines. End-to-end vs EstimatorQNN driven from Python (~100–400×), and kernel-level vs Aer's compiled statevector estimator (~1.7–3×, forward only) — so the win is not just deleted orchestration. Wall-time is shown until MLX hits the memory cliff (~22–26 qubits, single statevector).

Trains identically. Same circuit, init, data, and optimizer (SGD): the MLX layer (mx.grad) and the Qiskit QNN (qnn.backward) produce the same loss curve to ~1e-7 — same training, just faster.

Tests

uv run pytest

Covers gate/statevector correctness, gate unitarity and norm preservation, the little-endian convention, layer training, a finite-difference gradient check, and forward + weight-gradient + input-gradient parity with Qiskit across a random gate sweep (Qiskit-dependent tests skip automatically if Qiskit is absent).

Changelog

See CHANGELOG.md.

License

MIT — see LICENSE.