Unitary time evolution

Time evolution of quantum systems is always given byUnitary Transformations. If the state of a quantum system is|ψ〉, then at a later time

|ψ〉 → U |ψ〉.

Exactly what this operator U is will depend on theparticular system and the interactions that it undergoes. Itdoes not, however, depend on the state |ψ〉. Thismeans that time evolution of quantum systems is linear.

Because of this linearity, if a system is in state |ψ〉 or |φ〉or any linear combination, the time evolution is given bythe same operator:

(α|ψ〉+ β|φ〉) → U(α|ψ〉+ β|φ〉) = αU |ψ〉+ βU |φ〉.– p. 1/25

The Schrödinger equation

As we have seen, these unitary operators arise from theSchrodinger equation

d|ψ〉/dt = −iH(t)|ψ〉/~,

where H(t) = H†(t) is the Hamiltonian of the system.Because this is a linear equation, the time evolution mustbe a linear transformation. We can prove that this must be aunitary transformation very simply.

– p. 2/25

Suppose |ψ(t)〉 = U(t)|ψ(0)〉 for some matrix U(t) (which wedon’t yet assume to be unitary). Plugging this into theSchrödinger equation gives us:

dU(t)

dt= −iH(t)U(t)/~,

dU †(t)

dt= iU †(t)H(t)/~.

At t = 0, U(0) = I, so U †(0)U(0) = I. We see that

d

dt

(

U †(t)U(t))

=1

~U †(t)

(

iH(t)− iH(t))

U(t) = 0.

So U †(t)U(t) = I at all times t, and U(t) must always beunitary.

– p. 3/25

For time-independent Hamiltonians we can easily writedown the solution to the Schrödinger equation. Using thespectral theorem, we choose a basis {|k〉} of eigenvectors ofH with eigenvalues Ek, H|k〉 = Ek|k〉. We then write |ψ(t)〉in terms of this basis:

H =∑

k

Ek|k〉〈k|, |ψ(t)〉 =∑

k

αk(t)|k〉,∑

k

|αk(t)|2 = 1.

Knowing |ψ(t)〉 = exp(−iHt/~)|ψ(0)〉 for any t means know-ing the amplitudes αk(t). From the Schrödinger equation,

dαkdt

= −iEkαk/~ =⇒ αk(t) = exp(−iEkt/~)αk(0).

Each energy eigenstates undergoes a steady phase rotation.– p. 4/25

Bloch sphere rotation

Any 2× 2 Hermitian operator can be written H = aI + bX+

cY + dZ with real a, b, c, d. Spin-1/2 unitaries take the form

U(t) = exp(

−(it/~)(aI + bX + cY + dZ))

We now need to use a very useful and important fact. Forgeneral operators A and B, usually

exp(A) exp(B) 6= exp(A+ B).

The one exception to this is when [A, B] = 0. In this caseonly, exp(A) exp(B) = exp(A+ B).

– p. 5/25

Since the identity commutes with everything,

exp

(

− it~

(

aI + bX + cY + d)

)

= exp

(

− iat~I

)

exp

(

− it~

(

bX + cY + dZ)

)

= e−iat/~ exp(

−(it/~)(bX + cY + dZ))

.

Since an overall phase is meaningless, we can always seta = 0. (This is not just true for spin-1/2; one can add orsubtract a term aI to any Hamiltonian.)

– p. 6/25

The most general spin-1/2 Hamiltonian is therefore

H = bX + cY + dZ = E0~n · ~σ

whereE0 =

√

b2 + c2 + d2,

~n = (nx, ny, nz) = (b/E0, c/E0, d/E0),

with n2x + n2y + n2z = 1 and ~σ = (X, Y , Z). The unitary is

exp(−iHt/~) = cos(E0t/~)I − i sin(E0t/~)~n · ~σ.

In the Bloch sphere picture this corresponds to a rotationaround the axis ~n at a rate E0/~. This is the most generalunitary transformation possible for spin-1/2.

– p. 7/25

Controlling unitaries

We generalize from steady rotation by assuming we canturn the Hamiltonian on and off. By turning a Hamiltonianon for a particular length of time, we can “rotate” the stateby a particular angle. For a spin-1/2, this means we canperform unitary transformations of the form

U(θ) = cos(θ/2)I − i sin(θ/2)~n · ~σ.

We can do the same with more complicated systems. For aD-dimensional system with a Hamiltonian H havingeigenvalues Ek and eigenvectors |k〉, we can do the unitary

U(τ) =∑

k

exp(−iEkτ/~)|k〉〈k|

for any τ .– p. 8/25

Building up unitaries

Unfortunately, we cannot always choose the exact values ofthe eigenvalues Ek or the eigenvectors |k〉. These aregenerally given to us by nature. But we sometimes canincrease the range of our options by combining severaldifferent unitaries in a row. The important thing to rememberis that any product of unitary operators is also unitary:

U †U = V †V = I =⇒ (U V )†(U V ) = V †U †U V = I .

Suppose there are two different Hamiltonians we can turn on:H1 and H2. Then we can perform the unitaries

U1(τ) = exp(−iH1τ/~), U2(τ) = exp(−iH2τ/~).

But we can do much more than these!

– p. 9/25

We can also do the unitaries

U2(τ2)U1(τ1), and U2(τ3)U1(τ2)U2(τ1),

and U2(τn)U1(τn−1) · · · U2(τ2)U1(τ1).

Let’s see how this works for the spin-1/2. Suppose we canturn on Hamiltonians

H1 = ExX, H2 = EyY .

These produce unitaries U1(θ) and U2(θ) which correspond,in the Bloch sphere representation, to rotations by θ aboutthe X and Y axes, respectively.

– p. 10/25

There is a theorem in geometry that a rotation by anyangle θ around any axis ~n can be done by doing threerotations in a row around the X and Y axes:

R~n(θ) = RX(φ3)RY (φ2)RX(φ1)

for some φ1, φ2, φ3.

Since every 2× 2 unitary is equivalent to a Bloch sphererotation about some axis ~n, any 2× 2 unitary equals

UX(τ3)UY (τ2)UX(τ1)

for some τ1, τ2, τ3 (up to an overall phase).

When we talk about “turning on” and “turning off”Hamiltonians, what do we really mean?

– p. 11/25

Here is a picture to give some idea:

For many experimental systems, unitaries are effected byturning precisely-tuned lasers on and off for precise lengthsof time.

– p. 12/25

Tensor products of unitaries

We have seen that Hilbert spaces of composite systems arerepresented by tensor products of the Hilbert spaces of thecomponent systems:

H12 = H1 ⊗H2.

Therefore, unitaries on the joint system also act on thislarger Hilbert space. Suppose that the Hamiltonian of asystem is a sum of terms which act only on the individualsubsystems:

H = H1 ⊗ I + I ⊗ H2.

The two terms represent the Hamiltonian of the first andsecond subsystems, respectively. In this Hamiltonian, thetwo subsystems are isolated, and do not interact.

– p. 13/25

H1 ⊗ I and I ⊗ H2 commute, so

U(t) = exp(−iHt/~)= exp(−i(H1 ⊗ I + I ⊗ H2)t/~)

= exp(−iH1 ⊗ It/~) exp(−iI ⊗ H2t/~)

= exp(−iH1t/~)⊗ exp(−iH2t/~)

≡ U1(t)⊗ U2(t).

It is a tensor product of unitaries.

What if the Hamiltonian is not a sum of terms which acton the individual subsystems, but includes terms whichact on both subsystems?

– p. 14/25

Interactions and entanglement

If the Hamiltonian has the form

H = H1 ⊗ I + I ⊗ H2 + Hint,

the unitary U(t) = exp(−iHt/~) will not be a tensorproduct, in general. In this case, we say that the twosubsystems interact.

If a tensor-product unitary U1 ⊗ U2 acts on a productstate |ψ〉 ⊗ |φ〉, then it will remain a product state. If ageneral U acts on it, generally the state becomesentangled. Since most states are entangled, to producethem from initial product states requires unitaries whichare not products. We must have interactions between thesubsystems to produce general unitary transformations.

– p. 15/25

Let’s take an example for two spin-1/2s: Hint = EintZ ⊗ Z.This yields unitary transformations of the form

U(θ) = cos(θ/2)I − i sin(θ/2)Z ⊗ Z.

Suppose we have an initial product state |Ψ〉 = |ψ〉 ⊗ |φ〉

= α1β1| ↑↑〉+ α1β2| ↑↓〉+ α2β1| ↓↑〉+ α2β2| ↓↓〉.

When we transform it by U(θ) it becomes

U(θ)|Ψ〉 = e−iθ/2α1β1| ↑↑〉+ eiθ/2α1β2| ↑↓〉+eiθ/2α2β1| ↓↑〉+ e−iθ/2α2β2| ↓↓〉,

which is no longer a product state for θ 6= mπ/2. Theinteraction has produced entanglement.

– p. 16/25

The no-cloning theorem

The restriction of time evolution to unitary operatorsmeans that certain kinds of evolution are impossible. Oneimpossible task is quantum cloning.

Suppose we have a system in an unknown state |ψ〉,and we wish to copy it, i.e., to transform a second systemstarting in some standard state |0〉 into the same state|ψ〉. Is there a unitary U such that, for any state |ψ〉,

U (|ψ〉 ⊗ |0〉) = |ψ〉 ⊗ |ψ〉?

If this is true, then also for |φ〉 6= |ψ〉

U (|φ〉 ⊗ |0〉) = |φ〉 ⊗ |φ〉.

– p. 17/25

Consider now a superposition state |χ〉 = α|ψ〉+ β|φ〉.By linearity,

U(|χ〉 ⊗ |0〉) = U(α|ψ〉+ β|φ〉)⊗ |0〉= α|ψ〉 ⊗ |ψ〉+ β|φ〉 ⊗ |φ〉6= (α|ψ〉+ β|φ〉)⊗ (α|ψ〉+ β|φ〉),

so U(|χ〉 ⊗ |0〉) 6= |χ〉 ⊗ |χ〉, which is a contradiction.Therefore, no such U exists.

This is the famous no-cloning theorem: quantuminformation, unlike classical information, cannot be copied.

– p. 18/25

This simple result has many profound consequences.For one, the state |ψ〉 of a system is not an observable. Given aquantum system, there is no way to tell in what state |ψ〉it was prepared.

If the state |ψ〉 is known, the state can be “copied” bypreparing another system. But it is impossible to copyan unknown quantum state.

This means that many techniques of classicalinformation theory (such as protecting information bymaking redundant copies, or having a fanout gate from asingle bit) are impossible in quantum information theory.

– p. 19/25

Quantum gates and circuits

We have seen that it is possible to build up new unitaryoperators by multiplying together some set of standardones. This is rather analogous to the situation in classicallogic, where any Boolean function can be built up from a setof standard functions of one or two bits, called logical gates:

We can similarly try to build up unitary transformations froma set of standard unitaries. We will call these quantum gates.

– p. 20/25

First we define the basic unit of quantum information:the quantum bit. This is the simplest possible quantumsystem, one with two distinguishable states. In otherwords, the quantum bit (or q-bit) is our old friend, thespin-1/2! We take the standard basis to be |0〉 ≡ | ↑Z〉,|1〉 ≡ | ↓Z〉.The simplest gate, affecting only a single q-bit, is theNOT gate:

|0〉 ↔ |1〉.

We see that this is also a familiar operator: X.

NOT is the only nontrivial one-bit classical gate. But inquantum mechanics, there are far more possibilities.

– p. 21/25

One important example with no classical analogue isthe Hadamard gate:

UH =1√2

(

1 1

1 −1

)

.

We write these unitaries with a convention similar tothat of classical logic gates:

The wires of the circuit diagrams are q-bits, and the gatesare unitary transformations acting on those qubits.(Such unitaries act on other q-bits as the identity.)

– p. 22/25

We can also define two-bit quantum gates. Oneexample is the controlled-not (CNOT):

UCNOT|00〉 = |00〉, UCNOT|01〉 = |01〉,UCNOT|10〉 = |11〉, UCNOT|11〉 = |10〉.

Here we write the unitary as a two-bit gate where |in〉and |out〉 are now two-bit states:

– p. 23/25

Quantum circuits

There are infinitely many possible two-bit gates; but inpractice such unitaries are difficult to do. Fortunately, itturns out that just the CNOT (or almost any other two-bitgate), together with one-bit gates, can be used to buildup any unitary. (We will prove this later in the class!)

When we combine standard unitary gates, we call theresulting unitary a quantum circuit. Here’s a simpleexample that uses three CNOT gates to swap the firstand second bits:

– p. 24/25

You can check that this this circuit does indeed swapthe two q-bits by acting with it on each of the basisstates for two q-bits:

|00〉 → |00〉, |01〉 → |10〉,|10〉 → |01〉, |11〉 → |11〉.

Note that the control and target bits can all be switched,and this will still be a swap gate.

A quantum circuit for a less trivial unitary will in generalbe much more complicated. The problem of designingquantum algorithms is largely the task of designing suchquantum circuits.

Next time: some simple examples with light.

– p. 25/25