3 Robust against Errors . . . [6P]

University of Regensburg WS 2019/20

Quantum Information Theory

Prof. John Schliemann Tue. H33 13pm c.t. &Thu. H34, 3pm c.t.

Dr. Paul Wenk Mon. 12pm c.t., H33

Sheet 10

1 Fidelity . . . [10P]

Thefidelity is defined asF(ρ1, ρ2) := tr q

ρ^1/2₁ ρ2ρ^1/2₁ ²

, whereρi are density matrices.

(a) (i) Given a qubit in an unknown state|ψicalculate the fidelity of a random guess |φi.

(ii) Now, show that the fidelityimproves if a measurement has been applied to the qubit. To see this, calculate theaverage fidelity of the guess after the measurement.

(b) Prove that the fidelity is invariant under unitary transformations, i.e.,

F(U ρ₁U^†, U ρ₁U^†) =F(ρ₁, ρ₂). (1)

2 Simple Error Correction . . . [10P]

Consider (as in the lecture) three qubits which are exposed to errors. This errors can be described by a linear combination (with coefficients lying inC) of the following unitary matrices,

{UCNOTU_CNOT⊗1₂, 1₂⊗U_N ⊗U_N, 1₂⊗U_P⊗U_P, 1₂⊗(U_PU_N)⊗(U_PU_N)}, (2) withU_N :=|0ih1|+|1ih0|,U_P :=|0ih0| − |1ih1|, andU_CNOT:=|0ih0| ⊗1₂+|1ih1| ⊗X, whereX =|0ih1|+|1ih0|.

Consider the three-qubit state

|Λi= (|00i+|11i)⊗ψ with ψ=α|0i+β|1i, |α|²+|β|²= 1, α, β ∈C. (3) How can an arbitrary error acting on Λ be corrected to regain the correct|ψi?

3 Robust against Errors . . . [6P]

Consider errors of the formE= exp(−iφσz/2) withφ∈R. Show that encoding|0iand|1isuch that

|0i 7→ |0i_c= 1

√2(|01i − |10i), (4)

|1i 7→ |1i_c= 1

√

2(|01i+|10i), (5)

makes an arbitrary superpositionα|0i_c+β|1i_c robust against errors of the formE⊗E.

4 Shor’s 9 qubit code . . . [8P]

(a) In the lecture the observables τ₁τ₂ and τ₂τ₃ with τ₁ = X₁X₂X₃, τ₂ = X₄X₅X₆, τ₃ = X₇X₈X₉ have been introduced. Show that the syndrome measurement for detecting phase flip errors in Shor’s code corresponds to measuring this observables.

(b) Construct a single qubit quantum operations model(ρ) =P

nE_nρE^†_n for quantum noise which replaces every ρwith a completly randomized state1/2. Even this can be corrected by Shor’s code!