---
kind: concept
slug: variational-quantum-circuit
title: Variational quantum circuit
summary: >-
  A small parameterised quantum circuit used as a trainable function. Bits encode the input, trained rotations and CNOT layers process it, and Pauli-Z expectations give the outputs. One card so far: a 4-qubit Q-network with 28 parameters, tested in simulation.
tags: [quantum-machine-learning, variational-circuit, nisq]
papers: ["1907.00397"]
related: [deep-q-learning, experience-replay]
status: draft
updated_at: 2026-10-24
---

## Overview

A variational quantum circuit (VQC) is a quantum circuit whose gate angles are
trained by a classical optimiser, much as neural-network weights are trained.
The input is written into the qubits, a fixed pattern of entangling gates and
trainable rotations follows, and the output is read as expectation values of
measurements [arxiv:1907.00397 §4.1]. The appeal is the small number of
trainable settings, and the circuit depth must stay short because current
noisy hardware makes errors grow with the number of gates [arxiv:1907.00397 §1]
[arxiv:1907.00397 §6.6].

The paper credits the circuit approach to Mitarai et al. (2018)
[arxiv:1907.00397 §1] [arxiv:1907.00397 §4]. That work has no card here, so
nothing from it is described on this page.

## Variants

Only one variant has a card. Its design choices are listed so that a later
card can be compared against them.

### Bit-encoded circuit Q-network (Chen et al., 2019)

The state index is written in binary, one bit per qubit. A bit set to 1 is
turned into the |1⟩ state with π rotations. Each layer applies a chain of CNOT
gates between neighbouring qubits, then one general single-qubit rotation with
three trainable angles on every qubit. The paper uses two layers
[arxiv:1907.00397 §4.1] [arxiv:1907.00397 fig. 5] [arxiv:1907.00397 eq. 15].
The output for each action is the Pauli-Z expectation of one qubit plus a
trainable bias, and the chosen action is the index of the largest output
[arxiv:1907.00397 §6.1]. The circuit serves as the Q-function inside deep
Q-learning [arxiv:1907.00397 §4] [arxiv:1907.00397 alg. 1]. Card:
[Variational Quantum Circuits for Deep Reinforcement Learning](/papers/1907.00397).

Reported results, all from simulation unless stated:

- Frozen lake uses 28 trainable parameters where a table has 64 entries, and
  the authors report a total reward of 0.9 after 198 episodes
  [arxiv:1907.00397 §6.4] [arxiv:1907.00397 §6.6].
- Cognitive radio with 2 to 5 channels reaches a total reward of 100 in
  simulation [arxiv:1907.00397 §6.4] [arxiv:1907.00397 fig. 8].
- Two short hardware runs, 7 and 5 episodes, are reported with no variance
  [arxiv:1907.00397 §6.5] [arxiv:1907.00397 tab. 3] [arxiv:1907.00397 tab. 4].

## When to use which

With one card there is nothing to compare against, so the table gives the
constraints under which the paper's evidence applies.

| Situation | Use | Why |
|---|---|---|
| Small discrete state space, a few bits per state, e.g. 16 states with 4 bits | This circuit Q-network | Frozen lake: 28 parameters against 64 table entries, 0.9 reward after 198 episodes [arxiv:1907.00397 §3.1] [arxiv:1907.00397 §6.4] [arxiv:1907.00397 §6.6] |
| More than 5 qubits | Not supported by the evidence | Simulation cost grows with qubit count and scaling beyond n ≤ 5 is untested [arxiv:1907.00397 §6.6] |
| Training on real hardware | Not supported by the evidence | Hardware runs are inference only, with the simulator-trained parameters [arxiv:1907.00397 §6.5] |
| Noisy execution | Simulated noise only | The noise study uses one 2019 device snapshot [arxiv:1907.00397 §6.3] [arxiv:1907.00397 app. Noise Information] |
| Speed or accuracy gain over a classical network | Not shown | The advantage the paper claims is a parameter count, not a speed or accuracy result [arxiv:1907.00397 §6.6] |

## Implementation sketch

The core is the read-out from a state to one value per action. The learning
loop around it is the one described on [deep-q-learning](/concepts/deep-q-learning).

```pseudocode
# Circuit read-out for one state  [arxiv:1907.00397 §4.1] [fig. 5]
# Input: state index, angles (2, n, 3), bias (n,); Output: n action values
def circuit_q_values(state_index, angles, bias):
    # Write the index in binary, one bit per qubit  [§4.1]
    bits = binary_digits(state_index, width=n_qubits)
    # Bit 1: pi rotations about x, then z  [eq. 7] [eq. 8]
    for qubit in range(n_qubits):
        if bits[qubit] == 1:
            rotate_x(pi, qubit)
            rotate_z(pi, qubit)
    # Two layers: CNOT chain, then one rotation per qubit  [eq. 15]
    for layer in range(2):
        for qubit in range(n_qubits - 1):
            cnot(control=qubit, target=qubit + 1)
        for qubit in range(n_qubits):
            general_rotation(angles[layer, qubit])
    # Pauli-Z expectation per qubit, plus a trainable bias  [§6.1]
    z = [expect_z(qubit) for qubit in range(n_qubits)]
    return [z[i] + bias[i] for i in range(n_qubits)]
```

## Contradictions

- Target parameters. Algorithm 1 computes the target value from the current
  parameters, while the text says a separate target network is used
  [arxiv:1907.00397 alg. 1] [arxiv:1907.00397 §2.3] [arxiv:1907.00397 §4].
- Convergence, noisy radio runs. The authors give an optimum at about 110
  iterations [arxiv:1907.00397 §6.4], and elsewhere "very close to 100 in
  fewer than 10 iterations" [arxiv:1907.00397 §6.6].
- Convergence, noiseless radio runs. The caption of the reward figure says
  "several iterations" for 2 and 5 channels, while the text says about 100
  [arxiv:1907.00397 fig. 8] [arxiv:1907.00397 §6.4].

## Pitfalls

- The released code does not match the paper in several places: the radio
  code uses 4 qubits for every channel count, and its exploration rule and
  target sync differ from the text. See the
  [card's Paper vs code section](/papers/1907.00397).
- Gamma is not stated in the paper. The released code uses 0.999, so the
  reported numbers are tied to that value. The frozen-lake episode length is
  also not stated.
- Weight initialisation is not stated in the paper, and the bias starting
  value is not stated either.
- The radio code reseeds its random generator from the clock, so its runs
  are not reproducible from the repository. It also uses an alias that NumPy
  1.24 removed, so it will not run on current NumPy without changes.
- Parameter counts are not performance. The 28-parameter figure is a
  comparison with a table, and the radio neural-network count comes from a
  cited work that was not checked [arxiv:1907.00397 §6.6].

## Papers

- [Variational Quantum Circuits for Deep Reinforcement Learning](/papers/1907.00397):
  a bit-encoded, two-layer circuit as the Q-function of deep Q-learning, tested
  on frozen lake and cognitive-radio channel selection in simulation.