Circuit Depth

The number of sequential time steps in a quantum circuit. A key metric for NISQ-era computation.

Circuit depth measures how many layers of gates must be applied sequentially. It’s one of the most important metrics for determining whether a circuit can run on real hardware.

Definition

Depth is the longest path through the circuit, counting layers of gates that cannot be parallelized:

Layer:   1    2    3    4
         │    │    │    │
q0: ──H──┼────●────┼────X──
         │    │    │    │
q1: ──H──●────┼────X────┼──
         │    │    │    │
q2: ─────⊕────⊕────H────M──

This circuit has depth 4.

Why Depth Matters

Decoherence

Qubits lose their quantum properties over time (T1, T2 times). Deeper circuits mean more time for errors to accumulate.

Rule of thumb: Circuit depth should be much less than $T_{2}$ / gate time.

Error Accumulation

Each gate has some error probability. Errors compound:

100 gates at 99.9% fidelity each: ~90% overall fidelity
1000 gates at 99.9% fidelity each: ~37% overall fidelity

NISQ Limitations

NISQ devices can typically run circuits with depth ~100-1000 before noise overwhelms the signal. Fault-tolerant systems will enable much deeper circuits.

Depth vs. Width Trade-offs

Sometimes you can trade depth for width (more qubits):

Approach	Depth	Width
Sequential	High	Low
Parallel	Low	High

Example: Computing $f (x_{1}) \land f (x_{2}) \land f (x_{3}) \land f (x_{4})$

Sequential: Depth 4, Width 1
Parallel tree: Depth 2, Width 4

Reducing Depth

Techniques for shallower circuits:

Parallelization: Run independent gates simultaneously
Gate cancellation: Remove redundant gate pairs
Approximate compilation: Trade accuracy for depth
Algorithmic redesign: Use algorithms with inherently lower depth

Two-Qubit Gate Depth

Often, two-qubit gate depth is more relevant than total depth, since two-qubit gates are typically slower and noisier:

1Q gates: Fast, high fidelity (~99.9%)
2Q gates: Slower, lower fidelity (~99-99.9%)

Reporting Depth

When comparing algorithms or hardware, always clarify:

Total depth vs. two-qubit depth
Native gate set used
Whether optimization was applied

See also: Quantum Circuit, NISQ, Decoherence, T2 Time

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