Problem Set # 10 Solutions Problem Set #10 (Due in Class, THURSDAY 02-Dec-10)

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MIT 2.111/8.411/6.898/18.435 Quantum Information Science I Fall, 2010 Sam OckoMASSACHUSETTS INSTITUTE OF TECHNOLOGY December 1, 2010 MIT 2.111/8.411/6.898/18.435 Quantum Information Science I November 22, 2010 Problem Set # 10 Solutions Problem Set #10 (due in class, THURSDAY 02-Dec-10)

P1: (Circuit models of quantum operations) Let ρout = k EkρinEk†. P1: (Circuit models of quantum operations) Let ρout = k Ek†ρinEk. (a) Give the operation elements Ek describing the mappingP for this circuit: (a) Give the operation elements Ek describing the mapping for this circuit:

What physical process does this describe? (b)What Give the physicalE for this process circuit, does assuming this describe?ρ = p 0 0 +(1 p) 1 1 : kMASSACHUSETTS INSTITUTEe | | OF− TECHNOLOGY| | MITρin 2.111/8.411/6.898/18.435ρout Answer: Quantum⊕ Information Science I We use the identity Ek =ρekENovemberU 0E ,22, and 2010 as a placeholder we say ψ = a 0 + b 1 . h | •| i | i | i | i What physical process does thisProblem describe? Set #10θ θ U(dueψ in0E class,= a THURSDAY00 + b cos( 02-Dec-10)) 10 + b sin( ) 01 P2: (Quantum noise and codes|)i Single | i qubit| quantumi · operations2 | i (ρ) model· 2 quantum| i noise which is corrected by quantum error correction codes. E θ 1 0 P1: ((a)Circuit Construct models operation of quantum elements operations for such) Let thatρ upon= inputE†ρ ofE . any state ρ replaces it with the 0E U ψ 0E = aE 0 + b cosout ( ) k1 k; in k E0 = θ completely randomizedh | | statei | iI/2.| Iti is amazing· 2 that| i even such noise models0 cos as( 2 ) this may be (a)corrected Give the by operation codes such elements as theEk Shordescribing code! the mapping for this circuit: (b) The action of the bit flip channel can be describedθ by the quantum operationθ (ρ)=(1 0 sin( 2 ) p)ρ + pXρX. Show1 thatE U thisψ may0E be= givenb sin an( alternate) 0 ; operator-sumE1 = representation,E as (ρ)=− h | | i | i · 2 | i 0 0 E (1 2p)ρ+2pP+ρP++2pP ρP where P+ and P are projectors onto the +1 and 1 eigenstates of − − − − − ThisX,(0 is+ called1 )/√ an2 and amplitude ( 0 1 ) damping/√2, respectively. channel, This as itlatter reduces representation the amplitude can be understood of a 1 state. as a| model | in which the| − qubit| is left alone with probability 1 2p, and is ‘measured’ by| i the environmentWhat physical in theprocess+ , doesbasis this describe? with probability 2p. − | |− (b)(b) Give Give the the E kforfor this this circuit, circuit, assuming assumingρ = pρ0e =0 p+(10 0p+) 1 (11 : p) 1 1 : P3: (Shor’s 9 qubit codek ) The Shor code is ablee to protect| | | againstih−| | phase |− flip| andih | bit flip errors on any qubit. ρin ρout ⊕ (a) Show that the syndrome measurement for detecting phase flip errors in the Shor code corresponds ρe to measuring the observables X1X2X3•X4X5X6 and X4X5X6X7X8X9. (b) ShowWhat that physical recovery process from does a phase this describe? flip on any of the first three qubits may be accomplished by Whatapplying physical the operator processZ1Z2 doesZ3. this describe? P2: (Quantum noise and codes) Single qubit quantum operations (ρ) model quantum noise which is P4: (Recentcorrected quantum by quantum communication error correction results codes. ) . Find a recentE paper in the literature about a re- centAnswer: (post-2008) theoteical or experimental advance in quantum communication, involving distributed qubits.(a) Construct Write a short operation (< 500 elements word) for summarysuch of that it, onupon the input QISwiki. of any See state instructionsρ replaces on it the with course the homepage,completelyhttp://web.mit.edu/2.111/ randomized state I/2.E It is amazing that even such noise models as this may be corrected by codes such as the Shor code! (b) The action of the bit flip channel can be described by the quantum operation (ρ)=(1 p)ρ + pXρX. Show that this may be given an alternate operator-sum representation,E as (ρ)=− E (1 2p)ρ+2pP+ρP++2pP ρP where P+ and P are projectors onto the +1 and 1 eigenstates of − − − − − X,(0 + 1 )/√2 and ( 0 1 )/√2, respectively. This latter representation can be understood as a| model | in which the| − qubit| is left alone with probability 1 2p, and is ‘measured’ by the environment in the + , basis with probability 2p. − | |− P3: (Shor’s 9 qubit code) The Shor code is able to protect against phase flip and bit flip errors on any qubit. (a) Show that the syndrome measurement for detecting phase flip errors in the Shor code corresponds to measuring the observables X1X2X3X4X5X6 and X4X5X6X7X8X9. (b) Show that recovery from a phase flip on any of the first three qubits may be accomplished by applying the operator Z1Z2Z3. P4: (Recent quantum communication results) . Find a recent paper in the literature about a recent (post-2008) theoteical or experimental advance in quantum communication, involving distributed qubits. Write a short (< 500 word) summary of it, on the QIS wiki. See instructions on the course homepage, http://web.mit.edu/2.111/ Id: hw2.tex,v 1.4 2009/02/09 04:31:40 ike Exp 2

The above circuit is equivalent to this one:

Which applies X to the first qubit with probability 1 p and leaves it alone with probability p. Therefore, − E = √p I,E = 1 p X 0 · 1 − · This is called a bit-flip channel, since it flips the qubitp with a certain probability. P2: (Quantum noise and codes) Single qubit quantum operations (ρ) model quantum noise which is corrected by quantum error correction codes. E (a) Construct operation elements for such that upon input of any state ρ replaces it with the completely randomized state I/E 2. It is amazing that even such noise models as this may be corrected by codes such as the Shor code! Answer: We apply I,X,Y,Z gates with equal probability; I X Y Z E = ,E = ,E = ,E = . 0 2 1 2 2 2 3 2 We may describe the density matrix of one qubit as ρ = I/2 + aX + bY + cZ. Since I commutes with all operators, this component is left alone. Since X,Y,Z commute with half of the Ek and anticommute with half, these components will go to zero in the outputted density matrix, for example: IXI + XXX + YXY + ZXZ X + X X X (X) = E XE† = = − − = 0, E k k 4 4 Xk and likewise for Y, Z. Therefore, by linearity for all inputs ρ: (ρ) = I/2. E (b) The action of the bit flip channel can be described by the quantum operation (ρ) = (1 p)ρ+pXρX. Show that this may be given an alternate operator-sum representation,E − as (ρ) = (1 2p)ρ + 2pP+ρP+ + 2pP ρP where P+ and P are projectors onto the E − − − − +1 and 1 eigenstates of X,( 0 + 1 )/√2 and ( 0 1 )/√2, respectively. This latter representation− can be understood| i | asi a model in| whichi − | thei qubit is left alone with probability 1 2p, and is ‘measured’ by the environment in the + , basis with probability 2p−. | i |−i Answer:

We use the identities X = P+ P , I = P+ + P . Doing some algebra, − − − (ρ) = (1 p)ρ + pXρX = (1 2p)ρ + p (XρX + IρI) = E − − (1 2p)ρ + p ((P+ P ) ρ (P+ P ) + (P+ + P ) ρ (P+ + P )) = − − − − − − − (1 2p)ρ + 2pP+ρP+ + 2pP ρP , − − − which is equivalent to measuring the state in the + , basis with probability 2p. | i |−i Id: hw2.tex,v 1.4 2009/02/09 04:31:40 ike Exp 3

P3: (Shor’s 9 qubit code) The Shor code is able to protect against phase flip and bit flip errors on any qubit. (a) Show that the syndrome measurement for detecting phase flip errors in the Shor code corresponds to measuring the observables X1X2X3X4X5X6 and X4X5X6X7X8X9. Answer:

The codespace of the Shor code is stabilized by X1X2X3X4X5X6 and X4X5X6X7X8X9. Any phase ﬂip operation Zj anti-commutes with one or both of these operators, and can thus be detected by measuring these observables. For example, if X X X X X X ψ = 4 5 6 7 8 9 | i ψ , then X4X5X6X7X8X9 (Z4 ψ ) = Z4 ψ . To measure an observable g in the Pauli group| i , we simply apply the circuit:| i − | i

We can verify that we measure 0 when g ψ = ψ , and 1 when g ψ = ψ . | i | i | i | i | i − | i (b) Show that recovery from a phase ﬂip on any of the ﬁrst three qubits may be accomplished by applying the operator Z1Z2Z3. Answer:

Since the codespace of the Shor code is stabilized by the operators Z1Z2, Z2Z3, and Z1Z3, applying a phase ﬂip operation of Z1, Z2, or Z3 to any state in the code space is equivalent to applying the operator Z1Z2Z3. Therefore, applying Z1Z2Z3 will reverse that operation. MASSACHUSETTS INSTITUTE OF TECHNOLOGY MIT 2.111/8.411/6.898/18.435 Quantum Information Science I November 4, 2010

Problem Set #10 (due in class, THURSDAY 02-Dec-10)

P1: (Circuit models of quantum operations) Let ρout = k EkρinEk†.

(a) Give the operation elements Ek describing the mappingP for this circuit:

What physical process does this describe? (b) Give the E for this circuit, assuming ρ = p 0 0 + (1 p) 1 1 : k e | ih | − | ih | What physical process does this describe? P2: (Quantum noise and codes) Single qubit quantum operations (ρ) model quantum noise which is corrected by quantum error correction codes. E (a) Construct operation elements for such that upon input of any state ρ replaces it with the completely randomized state I/E 2. It is amazing that even such noise models as this may be corrected by codes such as the Shor code! (b) The action of the bit flip channel can be described by the quantum operation (ρ) = (1 p)ρ+pXρX. Show that this may be given an alternate operator-sum representation,E − as (ρ) = (1 2p)ρ + 2pP+ρP+ + 2pP ρP where P+ and P are projectors onto the E − − − − +1 and 1 eigenstates of X,( 0 + 1 )/√2 and ( 0 1 )/√2, respectively. This latter representation− can be understood| i | asi a model in| whichi − | thei qubit is left alone with probability 1 2p, and is ‘measured’ by the environment in the + , basis with probability 2p−. | i |−i P3: (Shor’s 9 qubit code) The Shor code is able to protect against phase flip and bit flip errors on any qubit. (a) Show that the syndrome measurement for detecting phase flip errors in the Shor code corresponds to measuring the observables X1X2X3X4X5X6 and X4X5X6X7X8X9. (b) Show that recovery from a phase flip on any of the first three qubits may be accomplished by applying the operator Z1Z2Z3. P4: (Recent quantum communication results) . Find a recent paper in the literature about a recent (post-2008) theoteical or experimental advance in quantum communication, involving distributed qubits. Write a short (< 500 word) summary of it, on the QIS wiki. See instructions on the course homepage, http://web.mit.edu/2.111/