How are elementary quantum gates realised?

Question

When expressing computations in terms of a quantum circuit, one makes use of gates, that is, (typically) unitary evolutions.

In some sense, these are rather mysterious objects, in that they perform "magic" discrete operations on the states. They are essentially black boxes, whose inner workings are not often dealt with while studying quantum algorithms. However, that is not how quantum mechanics works: states evolve in a continuous fashion following Schrödinger's equation.

In other words, when talking about quantum gates and operations, one neglects the dynamic (that is, the Hamiltonian) realising said evolution, which is how the gates are actually implemented in experimental architectures.

One method is to decompose the gate in terms of elementary (in a given experimental architecture) ones. Is this the only way? What about such "elementary" gates? How are the dynamics implementing those typically found?

Answer 1

Generally speaking, a realization of a quantum gate involves coherent manipulation of a two-level system (but this is nothing new to you, maybe). For example, you can use two long-lived electronic states in a trapped atom (neutral or ionized in vacuo) and use an applied electric field to implement single-qubit operations (see trapped ions or optical lattices, for example).

Alternatively, there are solid-state solutions like superconducting qubits or silicon-defect qubits which are addressed by radio-frequency electronics. You can use microwave-addressed nuclear spin sublevels, or nitrogen vacancy cells in diamond. The commonality is that the manipulation and coupling of the qubits is via applied light fields, and there are a range of methods you can use to tune the level spacing in these systems to enable single-spin addressing or manipulate lifetimes.

The translation from the implementation to Hamiltonian is obviously dependent on your choice of system, but eventually it all boils back down to Pauli matrices in the end. The light field provides off-diagonal elements in your single-qubit operations, whereas two-qubit operations are trickier and techniques are very implementation-dependent.

Answer 2

I'll posit that, much as classical NAND and NOR gates can be generated with NMOS and PMOS transistors arranged in series and/or in parallel, quantum gates such as CCNOT (Toffoli) and CSWAP (Fredkin) and Hadamard gates can be generated with sums and/or differences of creation and annihilation operators.

Certainly this analogy that transistors are to classical gates as ladder operators are to quantum gates is not new; indeed, this appears to be at much of the heart of the intuition in Feynman's Quantum Mechanical Computers. Clearly CMOS technology is dominant nowadays while Feynman's paper illustrated a NAND gate with NMOS technology only.

By the time Lloyd envisioned his quantum mechanical simulator in 1993 he was focusing on electromagnetic pulses inducing CSWAP gates in molecules or quantum dot arrays or nuclear spins, and wanted to move away from designer Hamiltonians of the previous literature.

(This answer may be precisely against the motivation of the question and still too abstract, but @NeildeBeaudrap's comments made me think of transistors and of the connection to Feynman's paper.)