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What I get so far:

Analog quantum computing: The Hamiltonian is implemented on the QC, solution is found by e.g. quantum annealing. The whole state is changing continuously.

Digital quantum computing: We use gates and change the state or parts of the state in sequence.

Digital QC is more powerful.

Is this correct?

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  • $\begingroup$ You probably need to be more precise about your definition of analog quantum computing. If you allow time evolution by Hamiltonians, then that has the same power as a digital quantum computer. Quantum annealing, or specific variants thereof, may have in-built restrictions that cause them to be less powerful (not "universal"). $\endgroup$
    – DaftWullie
    Commented May 27, 2022 at 12:42
  • $\begingroup$ How can I implement a time evolution by Hamiltonians, simply by acting longer on the state? $\endgroup$
    – nuemlouno
    Commented May 27, 2022 at 15:00
  • $\begingroup$ Quantum annealing changes a Hamiltonian slowly such that the system always stays in the ground state. Gate-based quantum computing usually works by turning on some sort of Hamiltonian $\mathcal{H}$ for a time $t$ to evolve the state by some unitary $U = e^{i\mathcal{H} t}$ which realizes desired quantum gate. $\endgroup$
    – Chris E
    Commented May 28, 2022 at 4:01

2 Answers 2

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As someone else said, you're right depending on how you define "analog quantum computing" because it's a bit of an overloaded term.

Quantum annealers work by slowly changing a Hamiltonian that corresponds to the problem you wish to solve, and the ground state that's measured at the end of the process corresponds to solution. There are people who debate whether annealers are QCs, because the annealing process takes much longer than the coherence time of the qubits used, so they don't fully utilize the "quantum-ness" in terms of superposition and entanglement. Also, annealers are only useful for solving a few classes of problems.

Gate-based quantum computers can implement any classical computing process and can utilize superposition and entanglement to solve some problems substantially faster than is possible classically. Since they utilize these quantum resources and are more general machines which can solve more classes of problems, quantum computing is more powerful than quantum annealing.

However, this kind of quantum computing can be digital or analog. Most of the well-known algorithms (Shor's, Grover's, etc) are designed for digital quantum computing, where we mostly think about the qubits as superpositions of $|0\rangle$ and $|1\rangle$. However, there is also ongoing research and some early demonstrations of analog quantum computing where the total output state after some evolution corresponds to some wavefunction in a different system of interest. This is often called quantum simulation, and is a promising technique for quantum chemistry and understanding how electrons are distributed in different molecules. Both of these modes are powerful, but are designed different applications.

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  • $\begingroup$ Any reference on what you refer to as quantum simulation (analo qc)? $\endgroup$
    – Marion
    Commented Oct 12, 2022 at 22:24
  • $\begingroup$ Peter Zoller's group has done a good bit and they have a recent perspective on the subject here. Disclaimer: I haven't read it $\endgroup$
    – Chris E
    Commented Oct 14, 2022 at 14:55
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It depends on what you mean by 'powerful'.

Digital quantum computing is a framework in which you can implement any type of quantum computation, including the ones implemented by analog quantum computing. In that sense, digital is more 'powerful' than analog.

On the other hand, analog QC is usually easier to implement on the hardware. In that sense, analog is more 'powerful' than digital.

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