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Measurement & quantum computing • Teleportation (continuous variables) • LOQC (KLM (msmt-based computation)) • cluster-state computation (ditto) 6 Mar 2012
Measurement & quantum computing
• Teleportation (continuous variables)• LOQC (KLM (msmt-based computation))• cluster-state computation (ditto)
6 Mar 2012
Can you teleport continuous-variable states?
1
Recall Quantum Teleportation
(And the other three results just leave Bob with a unitary operation to do)
Bennett et al., Phys. Rev. Lett. 70, 1895 (1993)
BSM
If BSM finds A & S in a singlet state, then we know they have opposite polarisation.Let Bob know the result. If S and I were opposite,
and A and S were opposite,then I = A!
singletstates
S and I haveopposite polarisations
S I
Alice BobA (unknown state)
A good excuse for a junket!(teleportation over 144 km?)
How to measure the continuousanalog of Bell states ?
E1 + E2E1
E2 E1 – E2
X1 + X2;
P1 + P2
X1 – X2;
P1 – P2
We wish to learn about the “relative” state of two systems,without measuring the exact state of either...
Do homodyne measurement on the outcomes, to measuredifferences or sums of the chosen quadratures.(At best, one difference and one sum.)
How to generate the continuousanalog of Bell pairs?
X1
P1
X1 is well known
X2
P2
P2 is well known
E1 + E2 = E3E1
E2 E1 – E2 = E4
X3 + X4 = X1 is well known.P3 – P4 =P2 is well known.
Continuous-variable teleportation
"Unconditional quantum teleportation," A. Furusawa, J. Sorensen, S. L.Braunstein, C. Fuchs, H. J. Kimble, and E. S. Polzik, Science 282, 706 (1998).
LOQC((efficient) Linear Optical Quantum Computing)
2
How to build a quantum computer?Photons don't interact
(good for transmission; bad for computation)
Try: atoms, ions, molecules, Josephson Junctions,...or just be clever with photons!
Nonlinear optics: photon-photon interactionsGenerally exceedingly weak.
Potential solutions:Cavity QEDBetter materials (1010 times better?!)Measurement as nonlinearity (KLM-LOQC, cluster, QND)Novel effects (slow light, EIT, etc)
Photon-exchange effects (à la Franson)Interferometrically-enhanced nonlinearity
Controlled Phase Gate
H
One entangling operation is as good as another for QI.Controlled phase (do a Z-rotation on the target if the controlis 1) is as good as controlled NOT (do an X-rotation on the targetif the control is 1).
–1111
+1010
+0101
+0000Out (Zc, Zt)In (Zc, Zt)
1011
1110
0101
0000Out (Zc, Xt)In (Zc, Xt)
=Z
X=
H
0 or 10
“+”means 0+1
+ or –
+0 0
or1
+or–
But CNOT has a clear control/target,while CPHASE is symmetric...
• Isn’t there always one bit left unchanged?
“–”means 0–1
special |ψi >
a|0> + b|1> + c|2> a|0> + b|1> – c|2>
The dream of optical quantum computing
INPUT STATE
ANCILLA TRIGGER (postselection)
OUTPUT STATE
particular |ψf >
MAGIC MIRROR:Acts differently if there are 2 photons or only 1.In other words, can be a “transistor,” or “switch,”or “quantum logic gate”...
But real nonlinear interactions are typically 1010 times too weak to do this!
What can one do with purely "linear" optics?
Hong-Ou-Mandel as interaction?
|H>
a|H>+b|V>
If I detect a "trigger"photon here...
...then anything whichcomes out here musthave the opposite polarisation.
Two non-interacting photons became entangled, not only by meeting ata beam-splitter, but by being found on opposite sides (postselection).Choosing the state of one can determine which states of the other are allowedto be reflected (if we only pay attention to cases where coincidences occur.)
|1>
a|0> + b|1> + c|2> a'|0> + b'|1> + c'|2>
The germ of the KLM ideaINPUT STATE
ANCILLA TRIGGER (postselection)
OUTPUT STATE
|1>In particular: with a similar but somewhat more complicatedsetup, one can engineer
a |0> + b |1> + c |2> → a |0> + b |1> – c |2> ;effectively a huge self-phase modulation (π per photon).More surprisingly, one can efficiently use this for scalable QC.
KLM Nature 409, 46, (2001); and others since...
special |ψi >
a|0> + b|1> + c|2> a|0> + b|1> – c|2>
Measurement as a tool: KLM...INPUT STATE
ANCILLA TRIGGER (postselection)
OUTPUT STATE
particular |ψf >
Knill, Laflamme, Milburn Nature 409, 46, (2001); and others after.
MAGIC MIRROR:Acts differently if there are 2 photons or only 1.In other words, can be a “transistor,” or “switch,”or “quantum logic gate”...
special |ψi >
a|0> + b|1> + c|2> a|0> + b|1> – c|2>
Measurement as a tool: KLM...INPUT STATE
ANCILLA TRIGGER (postselection)
OUTPUT STATE
particular |ψf >
Knill, Laflamme, Milburn Nature 409, 46, (2001); and others after.
special |ψi >
a|0> + b|1> + c|2> a|0> + b|1> – c|2>
Measurement as a tool: KLM...INPUT STATE
ANCILLA TRIGGER (postselection)
OUTPUT STATE
particular |ψf >
Knill, Laflamme, Milburn Nature 409, 46, (2001); and others after.
MAGIC MIRROR:Acts differently if there are 2 photons or only 1.In other words, can be a “transistor,” or “switch,”or “quantum logic gate”...
C0
C1
T0
T1
r2=1/3,t2= -2/3
A quantum-interferencecontrolled-phase gate
Theory: Ralph, Langford, Bell & White, PRA 65, 062324 (2002)Experiment: O’Brien, Pryde, White, Ralph, & Branning, Nature 426, 264 (2003)
See other early experiments: Gasparoni et al., PRL 93, 020504 (2004); Pittman et al., PRA 68, 032316 (2004).Other early theory includes Ralph et al. 65, 012314 (01); Pittman et al., PRL 88, 257902 (02); etc.
controlphoton
targetphoton
RH = 1/3RV = 1
A far more robust version– do all the interference in polarization; no alignment to worry about.
Langford et al. , PRL 95, 210504 (2005)
The cost of postselectionOf course, if each gate only “succeeds” some fraction p of the time...
the odds of an N-gate computer working scale as pN.
Exponential cost cancels exponential gain in quantum computing.
But, clever observation: gates “commute” with teleportation.
Perform the gates first, on “blank” registers (photons fromentangled pairs, which in some sense could be in any state at all),and save up the gates that worked [linear cost!]. Only now teleportthe input qubits into the already-successful gates!
Alternate picture: the gates generated some interesting entangledstates as a resource, and joint measurements with those states enablequantum computation –– this is more explicitly the idea of cluster-state (“one-way”) quantum computation.
Gottesmann & Chuang, Nature 402, 390 (1999)
Some referencesFOR QI in general...
Best to start withNielsen & Chuang’s Quantum Computation andQuantum Information (Cambridge U.P., 2000)
Technical papers...
Dense coding and teleportation:Bennett & Wiesner, PRL 69, 2881 (1992)Mattle et al., PRL 76, 4656 (1996)Benett et al., PRL 70, 1895 (1993)Bouwmeester et al., Nature 390, 575 (1997)Furusawa et al., Science 282, 706 (1998)
Error-correcting codes:Steane, Proc. Roy. Soc. Lond. A 452, 2551 (1996)Shor, PRA 52, 2493 (1995)Knill et al, quant-ph/020717
UPCOMING TOPICS...
Linear-optics quantum computation:Knill, Laflamme, & Milburn, Nature 409, 46 (2001)Gottesmann & Chuang, Nature 402, 390 (1999)Ralph, Langford, Bell, & White, PRA 65, 062324 (2002)O'Brien, Pryde, White, Ralph, & Branning, Nature 426, 264(2003)Langford et al., PRL 95, 210504 (2005)
Cluster-state quantum computation:Nielsen, "Universal quantum computation using only...",quant-ph/0108020Raussendorf & Briegel, "A one-way quantum computer",PRL 86, 5188 (2001)Raussendorf & Briegel, PRA 68, 022312 (2003)Aliferis & Leung, "Computation by measurements: aunifying picture", quant-ph/0404082Nielsen, "Cluster-state Quantum Computation", quant-ph/0504097Walther et al, Nature 434, 169 (2005)
Weak-nonlinearity optical quantum computation:Nemoto & Munro, PRL 93, 250502 (04)
