All qubits in a chip can interact directly

01

Why people believe this

In a quantum circuit diagram, any two qubits can be connected by a CNOT gate. It seems natural to assume the hardware implements this directly.

02

The correction

Real quantum hardware has a fixed connectivity graph — only physically adjacent qubits can interact directly. On IBM devices this is a heavy-hex lattice. On Google Sycamore it is a 2D grid. To apply a gate between non-adjacent qubits, the compiler must insert SWAP gates to move the qubit states physically closer. Each SWAP requires 3 CNOT gates and adds significant noise. This qubit routing overhead can multiply the effective circuit depth by 2-5x on typical circuits.

03

Visual demonstration

Hardware topology comparison — click each to explore

Ideal (fully connected)

Every qubit can interact with every other. Circuit compiles directly with no overhead.

IBM Heavy-Hex

Most qubits have 2 neighbors, junction qubits have 3. No qubit has 4 connections — chosen to reduce crosstalk and improve gate fidelity.

Google 2D Grid

Qubits arranged in a 2D lattice. Each connects to up to 4 neighbors. Better connectivity than heavy-hex but still requires routing.

Routing overhead: A SWAP gate requires 3 CNOT gates and introduces 3x the noise of a single two-qubit gate. On IBM heavy-hex, typical circuits require 2-5x more gates after compilation than the abstract circuit shows.

Real quantum chips have fixed wiring. A circuit diagram showing a CNOT between any two qubits is an abstraction — the compiler must route the operation through the physical topology, often inserting many extra gates.

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04

Simulator note

This concept requires either mathematical proof or hardware-scale experiments beyond what a browser simulator can demonstrate. See the research notes for the canonical references.

05

Research notes