Quantum Computers Final Exam

Quantum Computers Final Exam

January 6, 2004

Do any 4 of the 6 problems.

1. Fidelity

(a) A single qubit is in an unknown pure state |ψi, selected at ran- dom from an ensemble uniformly distributed over the Bloch sphere (p(θ, φ) = _4π¹ sin θ). We take a guess |ϕi that is also selected ran- domly on the Bloch sphere. The fidelity F is defined as

F ≡ |hψ|ϕi|², F ≡ tr (ρ_ψρ_ϕ) ,

for pure states and density operators respectively. Calculate the expected fidelity hF i of our guess by either equations. (Hint: this problem can be solved by either integrating over the Bloch sphere (expressing the states in polar coordinates) or finding the density operator (the expected state) for both states.)

(b) We perform a measurement of the spin of |ψi on the z-axis, that is, a measurement in the basis {|0i, |1i}, the resulting state is described by the density operator

ρ = |hψ|0i|²|0ih0| + |hψ|1i|²|1ih1|.

Calculate the expected fidelity between the density operators be- fore (ρ_ψ = |ψihψ|) and after measurement.

(c) Briefly explain the meaning of the difference between the two fi- delities calculated above.

2. Consider the two qubit state

|Φi_AB = 1

√2|0i_A 1

2|0i_B+

√3 2 |1i_B

!

+ 1

√2|1i_A

√3

2 |0i_B+1 2|1i_B

!

, (a) Compute the partial traces ρ = tr (|Φi hΦ|) and ρ =

(c) Find the Schmidt decomposition of |Φi_AB. (Hint: match each pair of eigenstates from the two density matrices that have the same eigenvalues, and use the square root of that eigenvalue as its coefficient.)

3. Describe the effect of decoherence using the Bloch sphere parametriza- tion of the density matrix. The effect of environment phase-damping on qubit A is

|0i_A|0i_E →^q1 − p|0i_A|0i_E +√

p|0i_A|1i_E,

|1i_A|0i_E →^q1 − p|1i_A|0i_E +√

p|1i_A|2i_E,

where the environment (in a 3-d Hilbert space) has standard basis {|0i_E, |1i_E, |2i_E}. The Kraus operators for the evolution of ρ_A is

M₀ =^q1 − pI, M₁ =√

p 1 0

0 0

!

, M₂ =√

p 0 0

0 1

!

, and the qubit A evolves to

ρ⁰_A= M₀ρ_AM₀+ M₁ρ_AM₁+ M₂ρ_AM₂. (a) Calculate ρ⁰_A, using ρ_A= a b

c d

!

.

(b) What would happen if qubit A is allowed to decohere further?

(compare ρ_A and ρ⁰_A)

(c) The density matrix of qubit A corresponding to the point ~p ≡ (p₁, p₂, p₃) where |~p| ≤ 1 is

ρA = 1

2(I + ~p · ~σ) = 1

2(I + p1σ1+ p2σ2+ p3σ3) .

What is the point ~p⁰ corresponding to the evolved density operator ρ⁰_A in terms of p₁, p₂, p₃?

(d) Which point in the Bloch ball would be reached asymptotically after further evolution?

4. To distinguish two non-orthogonal states |ϕ₁i_A and |ϕ₂i_A, we could prepare an ancillary system B and apply the unitary transformation U that act as

|ϕ₁i_A|βi_B → |ϕ₁i_A|β₁i_B,

|ϕ₂i_A|βi_B → |ϕ₂i_A|β₂i_B,

then the state of A can be determined without disturbing the state by measuring system B alone.

(a) Discuss the validity of this procedure. (Hint: Derive the relation between |β1iB and |β2iB from the unitarity of U)

(b) What if |ϕ₁i_A and |ϕ₂i_A are orthogonal?

5. Suppose there is a physical system that has two possible states: ρ₁ with probability p1 and ρ2 with probability p2; that is, its state can be interpreted as p₁ρ₁ + p₂ρ₂. We wish to perform a measurement that can determine its state with minimum error.

Suppose our measurement has two possible outcomes, projections E₁ and E₂ = I − E₁; that is, if we measure ρ, we would get the state ρE₁ with probability tr (ρE₁), and ρE₂ with probability tr (ρE₂). We guess that the state is ρ₁ when E₁ is measured and ρ₂ otherwise, then the probability of error is

perror = p1tr (ρ1E2) + p2tr (ρ2E1) .

(a) Show that if the eigenstates and eigenvalues of p₂ρ₂ − p₁ρ₁ is |ii and λ_i, that is,

p₂ρ₂− p₁ρ₁ =^X

i

λ_i|iihi|,

where the |ii’s form an orthonormal basis, then the error proba- bility is

p_error= p₁+^X

i

λ_ihi|E₁|ii.

(b) Find the non-negative operator E₁ that minimizes p_error, show that the resulting error probability from this projection is

(p_error)_optimal = p₁+ ^X λ_i.

Show that

(p_error)_optimal = 1 2 −1

2kp₂ρ₂− p₁ρ₁k_tr. (Hint: tr (p₂ρ₂− p₁ρ₁) = p₂− p₁ =^P_iλ_i.)

6. Use any definition of (p_error)_optimal from the previous problem:

(a) Calculate (p_error)_optimal for ρ₁ = ρ₂ = ρ. (Hint: consider the cases where p₁ > p₂ and p₂ > p₁.)

(b) Calculate (p_error)_optimal for ρ₁ and ρ₂ with supports on orthogonal subspaces. That is ρ₁ρ₂ = 0 = ρ₂ρ₁ where 0 is the zero matrix.

(Hint: use the form of trace norm kρk_tr = tr

h

ρ^†ρⁱ

1 2



.)

(c) Briefly explain the reason for the value of (p_error)_optimal in the pre- vious two situations.

Pauli Matrices

σ1 = 0 1 1 0

!

, σ2 = 0 −i i 0

!

, σ3 = 1 0 0 −1

!

General Qubit State

Vector (Pure)

|ψ(a, b)i = a|0i + b|1i = cosθ

2|0i + e^iφsinθ

2|1i = |ψ(θ, φ)i

|0i ≡ | ↑zi, |1i ≡ | ↓zi Density Operator (Pure)

ρ(θ, φ) = |ψ(θ, φ)ihψ(θ, φ)| = 1

2(I+ˆn·~σ) = 1

2(I+sin θ cos φσ₁+sin θ sin φσ₂+cos θσ₃) ˆ

n = (sin θ cos φ, sin θ sin φ, cos θ) Density Operator (General)

ρ(~p) = 1

2(I + ~p · ~σ) = 1

2(I + p1σ1 + p2σ2+ p3σ3)

|~p| ≤ 1

Qubit Measurement

P (↑ˆn) = tr (| ↑nˆih↑nˆ |ρ) = tr

1

2(I + ˆn · ~σ)ρ



Partial Trace

Two General Systems

|ψiAB =^X

iµ

aiµ|iiA|µiB

ρ_A= tr_B(|ψi_{AB AB}hψ|) = ^X

ijµ

a_iµa^∗_jµ|ii_AAhj|

ρ_B = tr_A(|ψi_{AB AB}hψ|) =^X

iµν

a_iµa^∗_iν|µi_{B B}hν|

Two Qubits

|Ψi_AB = a|00i + b|01i + c|10i + d|11i

ρ_A = (|a|²+ |b|²)|0ih0| + (ac^∗+ bd^∗)|0ih1| + (a^∗c + b^∗d)|1ih0| + (|c|²+ |d|²)|1ih1|

ρ_B = (|a|²+ |c|²)|0ih0| + (ab^∗+ cd^∗)|0ih1| + (a^∗b + c^∗d)|1ih0| + (|b|²+ |d|²)|1ih1|