more measurement & computing (clusters, n00n states, zeno

More measurement & computing(Clusters, N00N states, Zeno gates,...)

13 Mar 2012

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)

“One-way” quantum computation(cluster or graph states)

(another type of measurement-induced computation)

1

Cluster state notationmeans: (1) prepare all qubits in 0+1

(2) implement CPHASE (CQ) on any pairs connected by edges.

HHH

000+001+010-011+100+101-110+111

2

1

3

1 2 3Z1X2Z3 = 1

Suppose we measure X2 = 1.Then Z1Z3=1, i.e., Z1 = Z3.

Otherwise, we find X2 = –1.Then Z1Z3 = –1, i.e., Z3 = –Z1,which we can fix with a spin flip.

We have “teleported” bit 1 to bit 3.

Line clusters as “wires”

G

c

t 4

XGZtZcZ4 = 1If I measure XG=1, then ZcZtZ4 = 1,so Z4=ZcZt; this is basically a CNOT.

Computing with clusters

c(in)

t(in)

c(out)

t(out)

Embedding into “circuits”:

Computing with clusters, big picture

Cluster-state QC is a parallel approachutilizing the power of measurement

(as is the QND approach of Munro, Nemoto, et al.)

Rausssendorf &Briegel, PRL 86, 5188 (01)Walther et al., Nature 434, 169 (05)

Optical circuit for generation of a4-photon cluster state.

“Amplifying” weak(-ish) nonlinearities by using them to make measurements,which in turn lead to gates: Nemoto & Munro, PRL 93, 250502 (04)

Another interesting use of entangled resources

2

Jon Dowling’s Slide of Magic BS

Oscillates N times as fast

Theory: H. Lee et al., Phys. Rev. A 65, 030101 (2002); J. Fiurásek, Phys. Rev. A 65, 053818 (2002)

˘

Highly number-entangled states(“3003” experiment).

Important factorisation:

=+

A "noon" state A really odd beast: one 0o photon,one 120o photon, and one 240o photon...but of course, you can't tell them apart,let alone combine them into one mode!

States such as |N,0> + |0,N> (“N00N” states) could revolutionize metrology (from atomic clocks to optical-interferometric sensing),and have been proposed for lithography as well.

But how to make them?

M.W. Mitchell et al., Nature 429, 161 (2004)

How does this get entangled?

H

H

2H

Non-unitary


H

V

H & V

Perfectly unitary


H

60˚

H & 60˚

Non-unitary(Initial states orthogonaldue to spatial mode; finalstates non-orthogonal.)

Trick #1

How to combine three non-orthogonal photons into one spatial mode?

Yes, it's that easy! If you see three photonsout one port, then they all went out that port.

"mode-mashing"

Post-selective nonlinearity

Trick #2

Okay, we don't even have single-photon sources*.

But we can produce pairs of photons in down-conversion, andvery weak coherent states from a laser, such that if we detectthree photons, we can be pretty sure we got only one from thelaser and only two from the down-conversion...

SPDC

laser

|0> + ε |2> + O(ε2)

|0> + α |1> + O(α2)

εα |3> + O(α3) + O(ε2) + terms with <3 photons

* But we’re working on it (collab. with Rich Mirin’s quantum-dot group at NIST)

Trick #3But how do you get the two down-converted photons to be at 120o to each other?

More post-selected (non-unitary) operations: if a 45o photon gets through apolarizer, it's no longer at 45o. If it gets through a partial polarizer, it could be anywhere...

(or nothing)

(or <2 photons)

(or nothing)

But do you really need non-unitarity?V

HPBS

Has this unitary, linear-optics, operation entangled the photons?• Is |HV> = |+ +> - |– –> an entangled state of two photons at all,

or “merely” an entangled state of two field modes?• Can the two indistinguishable photons be considered individual systems?• To the extent that they can, does bosonic symmetrization mean that

they were always entangled to begin with?

Is there any qualitative difference in the case of N>2 photons?

Where does the weirdness come from?

HHH+VVV

(not entangled bysome definition)

AB

C

If a photon winds up in each of modes {A,B,C}, then the three resultingphotons are in a GHZ state – 3 clearly entangled subsystems.You may claim that no entanglement was created by the BS’s and post-selection which created the 3003 state... but then must admit that some is created by the BS’s & postselection which split it apart.

Making triphoton states...

HV(H+V) ≈ R3 + R2L + RL2 + L3E.g.,

In HV basis, H2V + HV2 looks “number-squeezed”; in RL basis, phase-squeezed.

It works!

Singles:

Coincidences:

Triplecoincidences:

Triples (bgsubtracted):

M.W. Mitchell, J.S. Lundeen, and A.M. Steinberg, Nature 429, 161 (2004)

Another entangling gate: root-SWAP

111110 + i 011001 + i 10010000OutIn

Double wells as entanglers

2? What’s 2?!

A two-particle double-well

But cf. the fascinating “exchange gates” ofAnderlini et al., Nature 448, 452 (2007)!!

This is just like the HOM!A swap is just a mirror, or a Mach-Zehnder interferometer;a root(SWAP) is a beam splitter... or half a M-Z interferometer?

in swapped

entangled? if you post-select coincidences, yes.a valid logical state of 2 qubits? Not if both photons end up in the same arm...How to avoid illegal states, and postselection?

Some more references(incomplete!)Clusters

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: a unifying picture", quant-ph/0404082Nielsen, "Cluster-state Quantum Computation", quant-ph/0504097Walther et al, Nature 434, 169 (2005)Briegel et al, Nature Physics 19 (2009) {review of msmt-based computation}Anderlini et al, Nature 448, 452 (2007) {the “exchange-interaction” gate}Nemoto & Munro, PRL 93, 250502 (04) {weak nonlinearities / QND gate}

Zeno gate:Franson, Jacobs, and Pittman, PRA 70, 062302 (2004)

{Zeno gate}U.S. Patent 6995404 (www.patentstorm.us/patents/6995404/description.html)

N00N states & generation:H. Lee et al., Phys. Rev. A 65, 030101 (2002)J. Fiurásek, Phys. Rev. A 65, 053818 (2002)M.W. Mitchell et al., Nature 429, 161 (2004)

more measurement & computing (clusters, n00n states, zeno

Documents