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From Quantum Circuits, there are two statements that are not clear.

Quantum Circuits Quantum circuits are collections of quantum gates interconnected by quantum wires. The actual structure of a quantum circuit, the number and the types of gates, as well as the interconnection scheme are dictated by the unitary transformation, U, carried out by the circuit. Though in our description of quantum circuits we use the concepts input and output registers of qubits, we should be aware that physically, the input and the output of a quantum circuit are not separated as their classical counterparts are; this convention allows us to describe the effect of unitary transformation carried out by the circuit in a more coherent fashion. In all descriptions of quantum circuits in addition to gates, we see quantum wires that move qubits and allow us to compose more complex circuits from simpler ones that, in turn, are composed of quantum gates. We compose components by connecting the output of one to the input of another; we also compose operations when the results of an operation are used as input to another. The composition does not affect the quantum states. The quantum wires do not perform any transformations in a computational sense; sometimes we can view them as transformations carried out by the Pauli identity operator ∑I.

What does

  1. "the output of a quantum circuit are not separated as their classical counterparts are; " mean? What does it mean are not separated as their classical counterparts? In classical, we have transistors acting as gates - so aren't those connected too?

  2. "we see quantum wires that move qubits and allow us to compose more complex circuits from simpler ones that, in turn, are composed of quantum gates" mean. What are these wires physically in a quantum computer? I understand that we read quantum circuits from left to right and that they are unitary transformations. Are the wires just a way to show this flow is from left-to-right?

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For most qubit modalities quantum wires are best thought of as time while quantum gates are best thought of as electromagnetic pulses applied to individual qubits. This is in contrast to classical gates, where classical wires are etchings of metal inside a substrate while classical gates are, as you say, transistors arranged in series and/or in parallel.

Take, for example, a NOT gate (which negates a bit). Classically this is instantiated with an etching of an input wire $a$, forked to go into a PMOS transistor in series with an NMOS transistor, with the output wire $\bar a$ at that point of connection. We can separately probe the input wire $a$ independently from the output wire $\bar a$.

But this is in contrast with a Pauli $X$ gate (which similarly negates a qubit). To apply the $X$ gate to a qubit $|\psi\rangle$ at a particular depth of your circuit, you wait until the depth of note and then you apply an electromagnetic pulse (e.g., a laser) to your qubit as $X|\psi\rangle$. But the input wire is just time up to the $X$ gate, which can't be separately probed independently of the output wire.

I also really like the analogy that quantum circuits are akin to musical writing. Time marches left to right; the single notes and chords correspond to single-qubit gates and multi-qubit gates, respectively, and there's no feedback or fan-out/fan-in. Whereas a classical circuit has such fan-in and fan-out, and often utilizes feedback - for example to design flip-flops, a quantum circuit cannot effectuate the same kind of feedback.

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  • $\begingroup$ Thank you for those analogies, they are very helpful in understanding. $\endgroup$
    – Moo
    Oct 31, 2023 at 12:33

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