# Are operators unitary on a real quantum computer?

The question is more from the physical side of quantum computers. Can we say that operators are unitary or due to the NISQ nature, the operator (impact on particles) in reality deviates from this property? Links to primary sources are welcome.

A unitary operator represents a invertible operation that is fully devoid of noise. Similarly to how pure states represent us having perfect knowledge about the quantum state.

In realistic scenarios, this never happens. There is always some degree of noise and uncertainty that makes states mixed and operations non-unitary. In such situations it might be more appropriate to describe physical operations as suitable quantum channels, which can also account for noise.

It's also worth noting however that any quantum channel can be pictured as a unitary evolution that entangles the state with an ancillary "environment" degree of freedom that we don't have access to. So in this sense, you might say that any channel "is" unitary. But this point becomes rather moot in practice because if you don't have a way to access such ancillary degrees of freedom it's more appropriate to just describe the evolution as non-unitary (though such unitary descriptions can be very useful for theoretical analyses).

• "There is always some degree of noise and uncertainty that makes states mixed and operations non-unitary" - but is it true that noised gate is not unitary? Is it physically possible? We can perturbe unitary matrix such that it stays unitary (rotation matrix with petrurbed angle as an example). Commented Jul 17 at 20:15
• @МаринаЛисниченко it becomes non-unitary when some information is lost. For example to the environment via thermal noise, fluctuations, etc.
– glS
Commented Jul 18 at 7:56
• now I have another question. If unitarity is not guaranteed, then how the cumulative probability distribution of the measured output state should be equal to 1? Commented Jul 18 at 9:31
• because you don't need unitary evolution to have meaningful output probabilities. You get it even in full generality with a quantum channel and POVM
– glS
Commented Jul 18 at 10:46

To a good approximation yes they are unitary. Otherwise experiments like Rabi oscillations would not work. To the extent that they aren't unitary... well, that slight non-unitarity falls under your gate error budget and can be corrected with error correction like all other gate errors. It's probably not the most important error source, but it's possible to quantify it.

Here is an example. You can build a quanutm computer out of photons, with qubits being stored in whether a photon is on one path or another path. You can bounce those superposed photons around with mirrors (e.g. as is done in a Mach-Zhender interferometer). But conservation of momentum means that when a photon bounces off a mirror, it pushes the mirror a teeny tiny little bit. Not enough to reliably separate the pushed and unpushed states, but a bit. When the photon's path is superposed, this partially entangles the position of the mirror on each path with the path of the photon. The mirror's position is constantly decohering, so any mirror bounce in your system ends up as a slight non-unitarity. But it's not large enough to matter. If you made your mirrors lighter and lighter, eventually this would become a relevant source of error limiting the fidelity of your gates.

There are similar effects in other control systems, where making the control object smaller can limit gate fidelity due to the control object becoming entangled with the qubits in the computer and then decohering. I've been told that in superconducting quantum computers this can set limits on how weak you can make the control pulses. If you make your X gate pulse too weak, the photons in the control pulse start ending up entangled with the value of the qubits in the computer, so the gate becomes highly non-unitary.

• the question seems to be asking about nisq computers, for which the first few sentences you wrote don't really apply. Commented Jul 17 at 19:13
• @forky40 Only the half-sentence about error correction is non-NISQ. Rabi oscillations a very NISQ. Error budgets are very NISQ. Quantifying error is very NISQ. Good approximations are very NISQ. Commented Jul 17 at 19:33