Time Evolution Operators

In Quantum Mechanics, we know that every system has many quantum states. As a passage of time new states are evolved. What we wish to do is to find a relation between these initial states and the evolved states. For this purpose we use a linear operator which evolves the states over time which is famously called the Time Evolution Operator. Let us see the representation of a time evolution operator.

Let us start with an initial state |ψ(t₀)> and let the final state be |ψ(t)> at time t. Let’s take a Unitary operator U(t,t₀) which is linear operator which depends on t and t₀. The final state is obtained by the Unitary operator and the initial state in such a way:

|ψ(t)> = U(t,t₀) |ψ(t₀)>

where t > t₀. Let’s see a simple case, when the time t=t₀. This means that the state isn’t changed and thus we can say that U(t₀,t₀)=I where I is the Identity matrix. We obtain the structure of the Unitary operator by inputting the final state |ψ(t)> into the time dependent Schrodinger equation which is:

where i= square root of -1, h is the Planck’s constant d/dt is the rate of change with respect to time, ψ being the wavefunction at time t and H is the hamiltonian operator.

Next we input the state |ψ(t)> from the Unitary operator equation above into the schrodinger equation and try to solve the Schrodinger equation.

And thus we are left with the equation:

This equation is a first order differentiation equation and solving it we get the value of unitary operator as:

Taking h=1 and t₀=0 the equation comes out to be:

Thus we see the evolution of the quantum state as:

|ψ(t)> = exp{-itH} |ψ(0)>

Now let us look at the time evolution of some of the famous logic gates.

For us to work on a Quantum Circuit we need to extract the matrix for the Unitary operators for these gates. For the conversion of the exp{-iθA} we use the Taylor-Maclaurin series which tells that:

Using this equation we can convert the expansion series for exp{iθ} into Cos(θ)+iSin(θ) as:

Using the beautiful property of the Pauli gates where if A is a Pauli gate and if we multiply it with itself, we get a I (Identity matrix).

And thus if we take exp{-iθA} , in the above equation we get I at places where we have even powers (element 1,3,5…) and A gate at the places where we have odd powers (element 2,4,6…) . Thus exp{-iθA} comes out to be:

Cos(θ)I-i.Sin(θ)A

as the terms at the odd places make up the expansion of cos function and the elements at the even places make up the expansion of sin function.

Single Qubit gates Z, X, Y:

The time evolution unitary operator for the Z gate is exp{-iθZ} where θ corresponds to time. This is also referred as Rᴢ(θ) which is rotation about the Z axis by an angle θ.

exp{-iθZ} = Cos(θ/2)I-i.Sin(θ/2)Z, where Z is the Pauli matrix.

where ϕ goes from [0,π].

Similarly :

where θ goes from [0,π].

Here we see that the time evolution of these single qubit gates correspond to the rotation about the respective axis.

Let us now look at the two qubit gates ZZ, XX and YY:

It seems these operators are commonly used in certain algorithms like QAOA to reduce the depth of the circuit and thus we also look at the time evolution of these important operators.

Again following the rule we used before we can write :

Rzz(ϕ) = exp{iZ⊗Zϕ} = cos(ϕ)I⊗I-sin(ϕ)Z⊗Z

We can implement this matrix for a two qubit state by applying a sequence of CX.(I⊗Rz(θ)).CX

We see that Rzz(θ) changes the global phase of the |00>,|01>,|10>,|11> states according to the above given matrix and thus creating a relative phase for the superposition of these states.

Similarly Rxx(θ) can be written as cos(ϕ)I⊗I-sin(ϕ)X⊗X.

And Ryy(ϕ) comes out to be cos(ϕ)I⊗I-sin(ϕ)Y⊗Y.

Obtaining Rxx(θ) and Ryy(θ) from Rzz(θ):

We can obtain these two matrices by actually diagonalizing the operators XX and YY. As we know that X=HZH and Y=SXS†. We can bring both the operators in Z form.

Let us see it for XX gate.

exp{-iX⊗Xθ} can be written as exp{-iHZH⊗HZHθ} which can be further simplified as exp{-iH⊗H.Z⊗Z.H⊗Hθ}.

Thus we can write Rxx(θ) as H⊗H.exp{-iZ⊗Zθ}.H⊗H.

Now it is easy to implement Rxx(θ) on a Quantum Circuit which can be done as : H⊗H.CX.(I⊗Rz(θ)).CX.H⊗H

Similarly we can get the Ryy(θ) by following Y=SXS† and thus getting it easily done by S⊗S.H⊗H.CX.(I⊗Rz(θ)).CX.H⊗H.S†⊗S†.

But why are the Rxx and Ryy operators so important?

Let’s see how the unitary evolution of these gates change the quantum states.

Let’s take states |00>,|01>,|10>,|11>.

Applying Rxx(θ) to these states.

Rxx(θ)|00> = cos(θ)|00>-isin(θ)|11>
Rxx(θ)|01> = cos(θ)|01>-isin(θ)|10>
Rxx(θ)|10> = cos(θ)|10>-isin(θ)|01>
Rxx(θ)|11> = cos(θ)|11>-isin(θ)|00>

We see that these gates helped in achieving entanglement of some form and we get maximal entanglement or the bells states when θ becomes π/4. This is lot advantageous to the trivial H.CX way of creating entanglement because that only creates bell states while these two operators create different types of entanglements.

So here we discussed the basics of Time evolution Operator and tried to create time evolution operators for some famous gates and saw their significance.

References:

• Time Evolution : https://en.wikipedia.org/wiki/Time_evolution
• Time evolution in QM http://physics.mq.edu.au/~jcresser/Phys301/Chapters/Chapter15.pdf
• Quantum Logic Gate : https://en.wikipedia.org/wiki/Quantum_logic_gate