# Does the order in which qubits are measured matter in quantum computing?

I’ll define my question more specifically:

Lets suppose we have an $n$ bit quantum computer, with qubits labelled $q_{1},…,q_{n}$, and let $b$ be any bit string of length $n$ (let’s assume we’ve labelled each qubits’ states $0$ and $1$ so that we obtain an $n$ bit bit string when we measure the qubits).

The question is, is the probability that we obtain $b$ when we measure each qubits’ state dependent on the order in which we measure the qubits? I have a feeling the probability would be independent of the order the qubits are measured, but am struggling to prove this.

• Consider the case where we take the qubits one light year apart, measure them in close succession, and then bring them back together – John Dvorak Dec 7 '16 at 23:13
• @JanDvorak How could measuring the qubits when they're 1 light year apart and then bringing them back together alter the probabilities (of obtaining each measurement outcome), from the case where qubits are close together (as they would be in a quantum computer)? – Alex Michael Dec 7 '16 at 23:32
• It's an axiom of quantum mechanics that space-like separated measurements commute (otherwise you'd be getting different results depending on your frame of reference). If your measurements can be space-like separated, they commute. – John Dvorak Dec 7 '16 at 23:34
• @JanDvorak This is a little above my understanding of Quantum Physics (I'm a Computer Scientist studying Quantum Computing :) ) – Alex Michael Dec 8 '16 at 1:26
• Have you tried proving this using the definitions? That's always a good start. – Yuval Filmus Dec 8 '16 at 7:16

## 1 Answer

No, the order doesn't matter.

Proofs

• Algebra. Take an input state $\sum_k c_k |k_0 k_1 k_2 ...\rangle$. Apply the definition of measurement from your textbook to it. Compute the expression for the probabilities and outputs of each case when measuring qubit 0 then qubit 1. Do the same for measuring qubit 1 then qubit 0. Notice that the two expressions are equal. Generalize.

• Circuit moves. Measurement is equivalent to a CNOT gate from the target qubit onto an ancilla qubit that you simply don't use for anything else: So if you think you can change the outcome statistics by reordering measurements, you should think you can do the same without involving measurements at all (until a simultaneous measurement of all qubits at the end of the circuit). All you should need is independent CNOTs. But clearly independent CNOTs can be re-ordered.

• Play. For me, the fact that ordering doesn't matter wasn't proven so much as experienced. I dragged gates around in Quirk and noticed that measurement is easily the most boring operation. All measurement does is throw away off-diagonal density matrix elements. Even for the target qubit, measurement doesn't change the computational basis probabilities.