# Circuit Depth

Circuit Depth

Circuit depth is the count of time steps needed to execute all the gates in a quantum circuit. One time step is the time it takes to execute one gate. If N gates can be executed in parallel, then only one time step is needed. But if N gates need to be executed sequentially, then N time steps are needed. The term applies whether using quantum computing or classical computing.

Although the definition seems straightforward, there is an interesting debate about the definition of quantum circuit depth in response to the question “Definitions of a quantum circuit's depth and connectivity” on Stack Exchange Quantum Computing.

## What is Circuit Depth

During the Noisy Intermediate-Scale Quantum (NISQ) era of quantum computing, circuit depth is an indicator of whether or not a circuit may return useful results. Depth implies more gates, and gates introduce errors. NISQ qubits also have limitedc oherence – the length of time they can maintain quantum information – and depth can push beyond those limits. Both errors and decoherence threaten to return non-useful results.

One example of why this is important is the quantum phase estimation algorithm, a significant algorithm with substantial depth and multitudinous gates. Beyond a handful of qubits, it can only return noise. The long-term solution for this is quantum error correction (QEC), which will extend coherence times and reduce error rates. Unfortunately, interim solutions can only help modestly. Circuit cutting depth may be helpful by trading depth for width, however this technique is not without its own challenges.

For more information, answers to the Stack Exchange Quantum Computing question “Why is depth complexity relevant?” delve into quantum gate complexity versus circuit depth complexity. They also address T-depth, which is depth related specifically to the T gate, because it is the slowest gate to implement.

## Significance of Circuit Depth in Classical Computing

Classical computers have Boolean circuits that consist of Boolean logic gates: AND, NAND, NOR, NOT, OR, XNOR, and XOR. The depth of such a circuit is the count of time steps from the first input to the last output. As defined above, gates that execute in parallel count as the same time step.

The significance of this metric is threefold:

• Depth contributes to the complexity analysis of the circuit.
• Depth affects the performance of a circuit; deeper circuits take longer to execute.
• Depth contributes to circuit size, which affects computational resource requirements.

These consequences are all relevant to quantum computing. Theoretically, quantum algorithms need to have less complexity than their classical counterparts. Experimentally, they need to be faster than their classical counterparts. And the most highly coveted quantum algorithms are those for which their classical counterparts have insufficient computational resources to run.

## Quantum Computing and Circuit Depth

In quantum computing, circuit depth is the count of time steps from the initialization of the qubits to the final measurement taken. All gates executed in parallel count as one time step, even though the execution times of gates can vary quite a bit. As noted earlier, the T gate is relatively slow, and T-depth is a metric in its own right. But mid-circuit measurements and classical logic, although not implemented in every circuit, are even slower.

The significance of this metric is also threefold:

• All gates have to execute within the coherence limits of the qubits.
• Execution has to be faster than the runtimes of classical algorithms.
• Because of abstraction, circuits are deeper at execution time than when they are built.

While depth is an indicator of higher gate counts, additional consideration is given specifically to multi-qubit CNOT gates. Deeper circuits tend to have higher CNOT counts, and CNOTs have much higher error rates than single-qubit gates.

Fortunately, QEC is progressing. As advancements continue into the detection and correction of errors, the NISQ era will eventually transition into the fault-tolerant quantum computing (FTQC)era. As logical error rates decline, the issue of circuit depth will eventually lose its relevance.

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