Quantum Circuits and Protocols

Alice reaches the protocol railway

In the previous chapter, Alice learned that two wires can share one joint quantum state. Now the rails lengthen into circuits. A circuit is not a picture of particles moving through pipes; it is an ordered recipe for state evolution, classical outcomes, and conditional actions.

Prerequisite recap

You should already know how H and CNOT create $|\Phi^+\rangle$, why measurement is probabilistic, and why ideal circuit gates are unitary. This chapter adds the conventions that let those pieces become protocols.

Circuit notation

Most circuit diagrams read left to right. Each horizontal wire tracks one qubit through time. A box is a gate. A filled control dot connected to a target denotes a controlled operation. A meter symbol denotes measurement, which produces a classical bit.

For a two-qubit circuit, $H$ on the first qubit is $H\otimes I$. A CNOT with first qubit controlling second qubit is a four-dimensional unitary. Their product prepares a Bell state:

No-cloning

A classical file can be copied. An arbitrary unknown quantum state cannot. Suppose a universal physical copier existed. Including any fixed blank register and environment in its description, its action would have to preserve linearity; in the ideal closed-system picture, write it as a unitary:

for every $|\psi\rangle$. It could copy $|0\rangle$ and $|1\rangle$:

Linearity would then require

But two copies of $|+\rangle$ would be

Those states differ. The contradiction is the no-cloning theorem. It does not stop you from preparing many known $|+\rangle$ states; it stops a universal copier for arbitrary unknown states.

Teleportation

Quantum teleportation transfers an unknown qubit state using a shared Bell pair, a Bell-basis measurement, two classical bits, and a correction. It does not move matter and it does not transmit faster than light.

If Alice wants to send $|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$ to Bob, they first share $|\Phi^+\rangle$. Alice performs a joint Bell measurement on her unknown qubit and her half of the Bell pair. The result is two classical bits. Bob applies one of four corrections: $I$, $X$, $Z$, or $XZ$. After that correction, Bob's qubit is $|\psi\rangle$.

One standard circuit implements the Bell measurement with CNOT from the input qubit to Alice's Bell-half, then H on the input, followed by Z-basis measurement of both. If those recorded bits are $(m_1,m_2)$ in that order, Bob can apply $X^{m_2}Z^{m_1}$. Other diagrams may label or order the two classical bits differently, so a correction table is meaningful only together with its convention. When both corrections are needed, writing $XZ$ or $ZX$ differs only by an unobservable global phase on the corrected state.

Superdense coding

Superdense coding reverses the resource emphasis. With a shared Bell pair already in place, Alice can encode two classical bits by applying one of $I$, $X$, $Z$, or $XZ$ to her qubit and sending that qubit to Bob. Bob performs a Bell-basis measurement to recover the two bits.

The protocol does not send two bits through an isolated qubit alone. It uses a previously shared entangled pair plus one transmitted qubit. The textbook statement assumes the Bell pair, gates, channel, and Bell measurement work ideally; a physical implementation must also account for entanglement distribution, loss, noise, and decoding errors.

Worked example: teleporting $|+\rangle$

Suppose the Bell-measurement record is $(m_1,m_2)=(1,0)$. Under the convention above, Bob's pre-correction state is $Z|+\rangle=|-\rangle$. Applying $Z$ returns $|+\rangle$. Alice never needs to learn the amplitudes of the input, and Bob cannot complete the recovery until the classical bits arrive.

Common misconceptions

Guided practice

For teleportation, make a four-row table for $(m_1,m_2)=00,01,10,11$ and evaluate $X^{m_2}Z^{m_1}$. Then make a second resource ledger for superdense coding. Record what is prepared before the message, what physically crosses the channel, what measurement Bob performs, and which resource is consumed.

Chapter assessment

You should be able to read a small circuit, explain why no-cloning follows from linearity, and describe teleportation or superdense coding with correct resource accounting.

Summary

Next step

You can now audit a protocol operation by operation and resource by resource. Chapter 7 uses those circuit skills to build oracle algorithms whose interference answers carefully stated computational questions.

References and further study

• Charles H. Bennett et al., "Teleporting an Unknown Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels".
• Michael A. Nielsen and Isaac L. Chuang, Quantum Computation and Quantum Information.
• IBM Quantum Learning, Quantum teleportation and Superdense coding.