# Reversal of time arrow on IBM Q

It is well known that a quantum computer is reversible. This means that it is possible to derive an input quantum state $$|\psi_0\rangle$$ from an output $$|\psi_1\rangle$$ of an algorithm described by a unitary matrix $$U$$ simply by applying transpose conjugate to $$U$$, i.e.

$$\begin{equation} |\psi_0\rangle = U^\dagger|\psi_1\rangle \end{equation}$$

In article Arrow of Time and its Reversal on IBM Quantum Computer an algorithm for a time reversal and going back to an input data $$|\psi_0\rangle$$ is proposed. Steps of the algorithm are following:

1. Apply a forward time unitary evolution $$U_\mathrm{nbit}|\psi_0\rangle = |\psi_1\rangle$$
2. Apply an operator $$U_\psi$$ to change $$|\psi_1\rangle$$ to $$|\psi_1^*\rangle$$, where the new state $$|\psi_1^*\rangle$$ is complex conjugate to $$|\psi_1\rangle$$
3. Apply an operator $$U_R$$ to get "time-reversed" state $$|R\psi_1\rangle$$
4. Finally, apply again $$U_\mathrm{nbit}$$ to obtain the input state $$|\psi_0\rangle$$

According to the paper, the algorithm described above simulates reversal of the time arrow. Or in other words, it simulates a random quantum fluctuation causing a time reversal.

Clearly, when the algorithm is run on a quantum computer, it returns back to initial state but without application of an inverse to each algorithm step. The algorithm simply goes forward.

My questions are these:

1. Why it is not possible to say that an application of $$U^\dagger$$ on output of algorithm $$U$$ is reversal of time arrow in general case?
2. It is true that above described algorithm returns a quantum computer to an initial state but it seems that the algorithm simply goes forward. So where I can see the a reversal of time arrow?
3. The authors of the articles have found out that when a number of qubit involved in the time reversal algorithm is increasing, the effect of time reversal diminishes:

• How is it possible to reverse time for few qubits and concurently to preserve flowing of time in forward direction for another qubits?

• Does this mean that time flows differently for different qubits?

• When do the qubits return to commnon time frame to be possible to use them in another calculation?

• The arrow of time is a statistical concept (a law of large numbers really). It's not a meaningful notion at the level of qubits. See Emilio Pisanty's answer here. 2 and 3 are non-questions in the sense that they're based on faulty premises and misunderstandings of the notion of the time arrow as a thermodynamic concept. – Sanchayan Dutta Jan 4 at 7:46

Of course if we have unitary evolution $$|\psi_1\rangle = U|\psi_0\rangle$$ then $$|\psi_0\rangle = U^\dagger|\psi_1\rangle$$
I did not read the paper, but evidently the authors do something different, based on the following: the Schrödinger equation $$i\hbar\frac{\partial\Psi}{\partial t}=\hat{H}\Psi$$ changes its form if we substitute $$t\rightarrow -t$$ to complex conjugate: $$-i\hbar\frac{\partial\Psi}{\partial t}=\hat{H}\Psi$$ and its solution is also complex conjugate.
So time reversal is antiunitary operator $$U_RK$$ where $$U_R$$ is unitary and $$K$$ is complex conjugation.