# Survival probability quantum circuit

Suppose say that I have a quantum state $$\vert\psi\rangle$$ at time $$t = 0$$, which is now evolved by a hamiltonian $$H$$

$$e^{-iHt}\vert\psi\rangle$$. I can ask the question, how much of initial state is still available in the evolved state, this is called survival probability.

$$P(t) = \vert\langle \psi\vert e^{-iHt}\vert\psi\rangle\vert^2$$. Is there a circuit implementation for this survival probability? Any idea how to implement this or source to any link which discusses about this circuit implementation is appreciable.

• One may approximate the time evolution operator $e^{-iHt}$ using a quantum circuit, however, further assumptions on $H$ are usually needed. The keyword is "Hamiltonian simulation". If you're interested in a specific (type of) Hamiltonian, I recommend that you specialize your question such that a concrete answer is possible. Otherwise, I suggest that you read up a bit on the literature. arxiv.org/abs/2101.07808 contains a decent overview over the available methods (but there is no review paper AFAIK) Commented Dec 6, 2022 at 9:50
• @MarkusHeinrich I was actually thinking of Trotterizing the evolution part (assuming the evolution is Trotterizable) and do a swap test between initial state and evolved state, performing a proper measurement on ancilla qubit, i can find the mod square of overlap between these states. But I don't know whether this will work. Just wondering, is there a more efficient way to find this probability. I am trying to simulate Rabi oscillations using quantum circuit, lets say. Commented Dec 6, 2022 at 10:01
• Is $\psi$ unknown or why do you want to do a swap test? You would also need two copies for each run. If $\psi$ is known and a "simple" state, I'd rather rotate the basis measurement accordingly. Anyway, without more details, you may not get a useful answer. Commented Dec 6, 2022 at 12:04

We can calculate this survival probability $$P(t) = |\langle \psi | e^{-i \mathcal{H}t} | \psi \rangle|^2$$ in two ways. The first corresponds to the so-called Hadamard test, as shown in the figure below,
where $$H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$$ is the Hadamard gate and $$|\psi \rangle$$ is some $$n$$-qubit state. The probability of measuring the ancillary qubit in state $$|0\rangle$$ is $$\frac{1 + |\langle \psi | e^{-i \mathcal{H}t} | \psi \rangle|^2}{2} \equiv \frac{1 + P(t)}{2}$$.
Alternatively, we can avoid the use of an ancillary qubit and the corresponding control of the time evolution provided that we have access to the circuit $$U$$ such that $$U |0\rangle = |\psi \rangle$$. In such case, we can simply use the following ancilla-free circuit
By measuring all $$n$$ qubits, the probability of measuring all qubits in state $$|0 \rangle$$ (i.e., retrieving the fiducial state) is $$|\langle 0 | U^{\dagger} e^{-i \mathcal{H} t} U | 0 \rangle|^2 = | \langle \psi | e^{-i \mathcal{H} t} | \psi \rangle |^2 \equiv P(t)$$.
Of course, the remaining question is how to implement the time-evolution operator $$e^{-i \mathcal{H}t}$$, which is the Hamiltonian Simulation problem mentioned by @MarkusHeinrich. The simplest option is Trotterization, but more sophisticated approaches such as truncated Taylor series with linear combination of unitaries method or qubitization are also available.