Does a quantum computer have a clock signal and if yes how big is it?

There's no straightforward equivalent of the concept of clock rate in quantum computing. Quantum computers are supposed to produce algorithmic speedups only for very specific categories of problems. In simple words, quantum algorithms can be represented by quantum circuits which are basically a sequence of quantum gates. To give you an idea of how quantum gates are applied, I'll quote Peter Shor's answer:

Consider an ion trap. The ions represent qubits by using one electronic state as a $|0rangle$ and another as a $|1 rangle$. A quantum gate is performed by applying a $4 times 4$ unitary matrix to two of these ions. This is done by shining a sequence of laser pulses on the ions. It's not a physical device into which two ions are input, in which they interact, and out of which the ions come with their states changed.

Such laser pulses are applied at certain intervals of time. You might consider that rate to be a clock cycle rate of some sort. However, you can't immediately map it to a classical notion of "how many instructions are performed per clock cycle?", as quantum instructions (i.e., the state evolution of qubits induced by a single layer of simple quantum gates) can't directly be compared to classical instructions (e.g., bit shifting). So, your statement: "If quantum computers are so much faster they must have a clock rate which is a multiple of this." isn't quite right. In some sense, they might even have lower clock rates (say, 100 MHz) but perform a greater number of effective (classical) instructions per clock cycle (i.e., per contemporary set of laser pulses). Note that this notion of clock cycle rate will be lower bounded by the decoherence times of the qubits.

More importantly, even here, just like in classical computing, the clock cycle rate isn't the only factor determining the performance. Furthermore, it wouldn't make sense to compare the clock cycle rates of different architectures, say ion trap quantum computers with superconducting quantum computers.

Quantum computing does not promise computational speed-ups due to faster clock rates. Rather, the speed-ups are algorithmic. This means that, to achieve the same task (for suitable tasks that allow for this speed-up), quantum computers would need a smaller number of operations to produce an answer. These speed-ups exist even if each "single operation" takes the same time in a quantum computer as it does classically.

Quantum computers can be seen as "quantum accelerators" attached to a classical computer.

The application of quantum gates is a problem of sending signals to the quantum device/accelerator.

The classical aspect of signal sending is clocked by the classical computer. The speed of the quantum machine is determined:

• relatively: to the worst case complexity of executing the algorithm on the classical machine instead of the quantum one (imagine using a CPU instead of the quantum accelerator)

• exactly: time needed to run the algorithm on the quantum accelerator

The latter depends on the "gate times" of the accelerator.

Therefore, a quantum computer is clocked in multiple ways:

• by the clock of the classical one controlling the quantum accelerator

• by the unit of time needed to apply the quantum gate on the accelerator

However, the clock speed does not play a significant role to why quantum machines seem for particular algorithms more powerful.

I would also like to bring up the classical concepts of combinatorial logic vs sequential logic.

Combinatorial logic is that which has no internal state, e.g. registers, memory etc. For this type of logic, there is not use for a clock: input comes in, and output goes out approximately instantaneously.

Sequencial logic is the opposite. Its internal state gets update with every clock tick.

Modern computers use a combination of both, which allows to describe a wide range of algorithms in a step by step fashion, storing memory between each instantaneous step.

In quantum computing however, it is my understanding that you cannot store the internal state of the circuit: one measurement is made, and it instantaneously gives a result essentially instantaneously.

Therefore, in that sense, a quantum computer is analogous to a combinatorial circuit, and there isn't much sense in having clocks in the quantum computer itself.

See also: Quantum circuits explain algorithms, why didn't classical circuits?