Introduction

In the measurement-based quantum computation (MBQC) model, computation works differently from the circuit model. Instead of applying a sequence of unitary gates to an input state, we first prepare a graph state that acts as a computational resource, with the input embedded in it. We then run the computation by performing a sequence of single-qubit measurements. To compensate for the randomness of quantum measurements, we apply Pauli corrections conditioned on earlier measurement outcomes, a mechanism known as feed-forward. By the end, the resource state has been “consumed”: only the output qubits, which now hold the result, remain.

Let’s see this principle in action on the small circuit below:

Hadamard gate circuit

The procedure has three steps:

  1. Prepare an ancilla qubit (qubit 1) in the state \(\ketplus\) and apply a \(CZ\) gate between it and the input qubit (qubit 0).

  2. Measure qubit 0 along the \(\Xaxis\) axis. The outcome \(s_0\) is \(0\) if the qubit is projected onto \(\ketplus\) and \(1\) if it is projected onto \(\ketminus\).

  3. Apply a Pauli \(X\) correction to qubit 1 if \(s_0 = 1\).

For an arbitrary input state \(\ket{\psi} = a\ket{0}_0 + b\ket{1}_0\), the computation unfolds as follows (up to a global factor):

\[\begin{split}\ket{\psi'}_1 &\sim X_1^{s_0} \, {}_0\langle \pm | \left( a\ket{0}_0 \ketplus_1 + b\ket{1}_0 \ketminus_1 \right) \\ &\sim X_1^{s_0} \begin{cases} a\ketplus_1 + b\ketminus_1 & \text{if projected onto } \ketplus \ (s_0 = 0), \\ a\ketplus_1 - b\ketminus_1 & \text{if projected onto } \ketminus \ (s_0 = 1), \end{cases} \\ &\sim a\ketplus_1 + b\ketminus_1 .\end{split}\]

Both measurement outcomes lead to the same final state. This is the whole point of feed-forward: when \(s_0 = 1\), the \(X\) correction flips the sign of the \(\ketminus\) component and “corrects” the randomness of the measurement. If you compare the input and output states, you will notice that this circuit implements a Hadamard gate, since \(H\ket{\psi} = a\ketplus + b\ketminus\). Note also that the result now lives on qubit 1. The information has been teleported from the input register to the output register as a side effect of the computation.

Our example is a circuit enriched with ancillas, mid-circuit measurements and classical feed-forward. MBQC generalizes this idea: it provides a language and tools for expressing computations independently of any circuit representation.

In the next tutorials, we will see that it is possible to express any quantum computation using just four primitives:

  • preparing ancilla qubits in the \(\ketplus\) state,

  • entangling qubits with \(CZ\) gates,

  • performing single-qubit measurements along an arbitrary angle on the \(\XYplane\) plane,

  • applying Pauli \(X\) and \(Z\) corrections conditioned on previous measurement outcomes.