Circuit Depth and Width

Two simple numbers summarise the structure of any quantum circuit: width and depth.

Width

The width of a circuit is the number of qubits it uses. A circuit operating on $n$ qubits has width $n$ . Width directly sets the memory cost — a width- $n$ circuit manipulates a $2^n$ -dimensional state vector. A circuit that introduces an ancilla qubit to assist a calculation has larger width than one that avoids it.

Depth

The depth of a circuit counts the minimum number of time-steps required when gates acting on disjoint qubits are allowed to execute in parallel. Concretely, you assign each gate to the earliest layer that does not already contain a gate on any of the same qubits. The depth is the index of the last layer needed.

Consider this three-qubit example:

\text{Layer 1:}\; H_0 \parallel H_1 \qquad \text{Layer 2:}\; \mathrm{CNOT}_{0 \to 1} \qquad \text{Layer 3:}\; H_0 \parallel \mathrm{CNOT}_{1 \to 2}

Here $H_0$ means the Hadamard on qubit 0 and $\mathrm{CNOT}_{c \to t}$ denotes a controlled-NOT with control $c$ and target $t$ . The symbol $\parallel$ shows that the two operands run in the same layer because they act on distinct qubits. The circuit uses 3 qubits, so its width is 3, and it fills 3 layers, so its depth is 3.

Why depth matters

A naive implementation of an $n$ -qubit algorithm might place every gate sequentially, giving depth equal to the total gate count. Recognising which gates commute — because they act on different qubits — lets a compiler collapse them into the same layer, reducing depth without changing the computation. In the example above, the five gates collapse to depth 3 rather than 5.

Try it

The circuit described in the starter code has 3 qubits and the three layers shown above. Return its depth as an integer.

Run your code to see the quantum state.