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So here's something that's been bothering me. Given the time evolution of the wavefunction can only be unitary or discontinuous as a process of the measurement. So let the observables for our Hamiltonian be position $\hat x$, momentum $\hat p$ and energy $\hat H$. Does this mean the only possible states I can prepare in the lab are:

$$|\phi_1 \rangle = |E_0 \rangle $$

or

$$|\phi_2 \rangle = U|x_0 \rangle $$

or

$$|\phi_3 \rangle = U|p_0 \rangle $$

where $U$ is the unitary operator at arbitrary $t$, $| x_0\rangle$ is an arbitrary position eigenket, $|p_0 \rangle$ is an arbitrary momentum eigenket and $|E_0 \rangle$ is an arbitrary energy eigenket? Am I correct in interpreting this as meaning that there are only certain quantum computations which can be performed with such a system (in the sense that there is limited initial data one might input)?


Cross-posted on physics.SE

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  • $\begingroup$ I'm confused why you're even conceiving of a quantum computation as operating on a space with positions and momentums in the first place. Quantum computers generally operate on qubits. You can have path-encoded qubits, but also spin qubits or polarization qubits or many others. Could you rephrase your question in terms of qubits? $\endgroup$ Jun 20, 2021 at 18:57
  • $\begingroup$ @CraigGidney Is there a difference between a qubit and a ket? Also now that you mention it I suspect for spin I can always rotate my axis (of measurement of spin) and I guess the same will be true for polarization qubits but not for path encoded qubits ... I'll edit the last question $\endgroup$ Jun 20, 2021 at 19:07
  • $\begingroup$ Yes, there's a difference. A qubit is two level system. A ket is a state vector. One is part of the definition of the state space, the other is picking out a state from that space. $\endgroup$ Jun 20, 2021 at 19:10
  • $\begingroup$ @CraigGidney sorry my bad I am terrible with definitions. My thoughts were qubits were solely defined by orthogonality ... And in which case I'm uncertain how to rephrase the question in terms of qubits. But I still think it's a valid question $\endgroup$ Jun 20, 2021 at 19:15
  • $\begingroup$ Also I do suspect if I manage to frame the question in terms of qubits the answer would become self evident $\endgroup$ Jun 20, 2021 at 19:22

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This question is often relegated to the hardware side of quantum computing but is very important. Quantum computing theory assumes access to things like qubits (2-level quantum systems) and unitary operations acting on those qubits, where the unitary operations may be noisy. Alternatively, it assumes access to a highly entangled state, such as in measurement-based quantum computing, or to Gaussian states, squeezing operations, and various measurement protocols in continuous-variable quantum computing. All of these theories fundamentally assume that there is some underlying set of states and quantum state transformations that can be programmed to do a quantum computation.

Your question matches most closely to continuous variable quantum computation, in which the Hilbert space looks like that of a quantum harmonic oscillator. Then, there are certain initial states that are easier to access: the ground state (ie the vacuum), which would be the state of the system if you got rid of as much energy as possible, and thermal states, which arise when the system equilibrates with a large external bath of a fixed temperature. The possible state transformations are indeed of the form $U|\psi\rangle$, where $U$ is in this case achieved through things like interferometers and nonlinear crystals, which can be combined in various ways so as to make an [almost] universal gate set ("almost" because the measurement process can also be relied on to enact more transformations). There is also the possibility of things like displacement operations, enacted by lasers, which again simply change the form of $U$. So the magic here is that there are enough different types of unitaries $U$, which do not all commute with each other, that can be combined in enough ways to generate a huge variety of states, even from a set of initial states as boring as being in the vacuum.

In terms of qubits, things are a little easier to describe mathematically. Somehow, one gets access to a Hilbert space of dimension two, as well as unitary operations that transform the state in that Hilbert space. It turns out that you only need access to [multiple copies of] two noncommuting unitaries in order to generate all of the possible transformations, such as using rotations on the Bloch sphere about two different axes. If the qubit is the spin of an atom with an applied external magnetic field, the ground state will have the spin pointing in the same direction as the magnetic field, so such a state can be prepared using a magnetic field. Then, the other unitaries can be applied by adding other magnetic fields or the like. If the qubit is in two levels of an atom, the ground state can also be prepared by removing as much energy as possible, and then laser pulses can be used to drive transissions between the two levels in order to achieve unitaries $U$. If the qubit is in the polarization degree of freedom of a single photon, a polarizer can prepare any chosen initial state (with some probability of success) and wave plates can be used to enact a variety of unitary transformations. For each architecture, there are different ways of preparing states and of performing unitaries.

All in all, having a few (or one) initial states and a variety of noncommuting unitaries that can be applied for different amounts of time leads to a huge number of possible transformations! Different transformations are easier to achieve with different physical systems, so the goal is always to be able to do at least a certain set of transformations that are sufficient to generate arbitrary ones. To then do universal quantum computation, we also need access to gates that can act on more than one qubit at a time, so that again becomes a different question in each physical platform. Often, the easiest states to prepare are ground states of some Hamiltonian. Since we can externally control parameters of Hamiltonians (changing interaction strengths, applied external fields, etc.), we can help guide systems toward a variety of initial states that are easier for manipulation later on.

Aside: noisy transformations are described by quantum channels that are more general than unitary operations, so quantum channels are also accessible for state preparation and transformation. In general, quantum computation prefers unitary operations, so the quantum channels are more useful for characterizing why and how a quantum computation may be imperfect.

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