The OneWay Quantum Computer Computation from Correlations

1
The One-Way Quantum Computer Computation from
Correlations
Munich, 2000-01

• Dan Browne, Robert Raussendorf, Hans Briegel

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The One-Way Quantum Computer (QCC)

• Standard approach to quantum computation
• Unitary operations on a register of qubits
  followed by measurement of the qubits
• Unitaries decomposed into CNOT plus rotations
  Quantum Circuit Model
• One-way Quantum Computer or Cluster-state
  Quantum Computer (QCC) a different approach

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Outline of QCC

• Special Entangled State (Cluster State / Graph
  State) generated on (at least 2-d) array of
  qubits.
• Individual qubits measured in particular bases
  and particular order.
• Generates quantum state
• On a particular n qubits which are then measured.
• Arbitrary U universal quantum compution
• (given sufficiently large cluster state)

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Direct Implementation of U
Single qubit measurements in particular bases,
and particular order(Some bases may depend upon
previous measurement outcomes.)
Initial multi-partyentangled resource state
Measure final qubits
Implement U directly with specific measurement
pattern
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Implement Quantum Circuit
ControlledPhaseGate
Arbitrary Rotation
R
q
q
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Summary of Talk

• Stabiliser Formalism for states and measurements
• Cluster States (Graph States)
• Example Gate CNOT
• Computation from correlations
• Connecting Gates
• Accounting for randomness of measurements
• Further examples

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Pauli group

• Pauli operators
• Introduced to describe spin
• Useful in many other contexts (including QIS)
• n-qubit Pauli Group
• n-qubit tensor of Pauli operators with
  pre-factors (1, i)

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Measurements on the Bloch Sphere

• We can choose any basis for a measurement,
  represented by an axis through centre of Bloch
  sphere.

(Image stolen from E. Knills webpage!)
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Stabiliser Formalism

• Important applications
• Quantum error-correcting codes
• Compact description of multiparty
  entangledstates
• Operator O is stabiliser of state ?i if
• Specifying multiple stabilisers can define a
  sub-space, or even a specific state.

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Example Bell States

• Maximally Entangled 2-qubit states have
  stabilisers
• X X and Z Z
• i.e. if
• X X ?i ?i and Z Z ?i ?i
• ?i(1/2)1/2 (00i11i)
• The stabilisers define ?i (up to a phase) and
  directly represent the quantum correlations in
  the state.

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Manipulation of Stabilizers

• A set of stabilizers can be manipulated into
  different forms as necessary, in two main ways
• a) multiplication given two stabilizers Oa and
  Ob for ?i one or the other can be replaced by
  the product Oa Ob to give a new complete set of
  stabilisers for ?i.
• b) rotation rotation operators may be
  generated from one stabilizer and applied to
  others (I will explain this on blackboard if
  there is time!).

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Cluster States / Graph States

• The initial entanglement resource for the
  one-way quantum computer is the cluster
  state.
• Entangled Multi-party state on a 2-, 3-d array of
  qubits.
• Generated by
• Preparation of all qubits in state i.
• Application of two-qubit unitary S (controlled
  phase gate) between neighbouring qubits.

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Cluster States
S
This is a controlled phase gate between
neighbouring qubits.
Note, interaction between different pairs of
qubits commute.
This can be implemented in an optical lattice
with cold controlled collisions, seeD. Jaksch et
al Phys. Rev. Lett. 82 1975 (2000) .
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Cluster States

• Examples
• In two qubits Bell State
• In three qubits GHZ state
• In general, Cluster States have no simple state
  vector representation (no. of terms increases
  exponentially in no. of qubits).
• However, stabiliser formalism provides an easy
  and compact description.

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Stabilizers for the Cluster State

• A cluster state on a given qubit array A is
  defined by the following stabilisers.
• 8 a2A where ngbr(a) represents all nearest
  neighbours of qubit a.
• ?a 2 0,1.
• Since the state is completely defined by the
  stabilizer eigenvalue equations, all of its
  properties can be calculated in terms of the
  stabilisers.
• For simplicity and w.l.o.g. let us set ?a0, 8 a.

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Aside Graph States

• A generalisation of the cluster states, graph
  states have stabilisers of the same form, except
  the qubits need no longer to form a square grid,
  next-neighbours are now defined by edges of a
  graph.

See e.g. D. Schlingemann quant-ph/0305170, M.
Hein, et al, quant-ph/0307130
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Measurement with Stabilisers

• In the stabiliser formalism the result of
  single-qubit measurements is easy to calculate.
• Given an measured observable O
• a) If O commutes with all but one of the
  stabilisers
• replace the non-commuting stabiliser with O (
  pre-factor 1 to indicate measurement result) !
  Set of stabilisers for the new state
• b) If O does not commute with any of the
  stabilisers manipulate set using multipication
  and rotation of elements until one of the new
  stabilisers does commute with O.
• Then carry out replacement as in a).

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CNOT

• Using these tricks its simple to show that
• generates correlations

(Assuming for now positive eigenvalue is always
detected)
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Computation from correlations 1

• Given a state ?i with stabiliser X U X Uy
• Measure observable X on first qubit, use
  stabilisers to show that
• In first case, state U i has been generated on
  the second qubit.
• The state ?i is a resource to create U i. Is
  it a general resource for U? Connectivity?

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Correlations for computation

• To be a general resource one must be able to
  connect these states connect gate measurement
  patterns on the QCC.
• If we possess resource states for unitaries V and
  W,
• How can these be combined to generate combined
  unitary W V on our input state i?

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Connecting Gates

• Theorem A
• The following three sets of stabilizers are
  equivalent

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Connecting Gates

• Consider two 1-qubit gates, after the central
  qubits have been measured.
• Stabilisers after measurement

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Connecting Gates

• Consider two 1-qubit gates, after the central
  qubits have been measured.
• Equivalent to

Using Theorem A
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Connecting Gates

• Introduce additional qubit prepared in i.
• And apply the next neighbour control phase
  operation S which created the cluster state.

S
S
c
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Connecting Gates

• This changes the stabilisers as follows

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Connecting Gates

• The state is equivalent to having implemented the
  measurement patterns for V and W directly on a
  larger initial cluster state.

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Connecting Gates

• Rearranging the stabilizers by multiplication
  gives two stabilisers which commute with X
  measurements on A, c and b

Making these measurements, gives (assuming
eigenvalue 1 is measured).
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Connecting Gates

• Using Theorem A again, we obtain,
• Thus the combined measurement patterns for V
  and W linked with X measurements on A, c, and b
  give a combined pattern for WV.

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N-qubit gates

• These results generalise to n-qubit gates. The
  measurement pattern which produces a state with
  stabiliser
• realises n-qubit gate U. The patterns can be
  connected to form quantum networks.

For k2 1,n.
In a sense, the qubits i can be said to form the
input ports of the gate and the qubits o form
the output ports of the gate.
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More Example Gates

• Simple Rotation Ux(?)
• Controlled phase Gate

Proof on board (if time).

Measurement of observable cos(?) X sin(?) Y



to counteract randomness of measurements!
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Accounting for randomness of measurement results

• Above, I assumed that whenever a measurement was
  made the 1 eigenstate was detected.
• However, in actuality, this is not the case! 50
  of the time, the -1 eigenstate is detected.
• This has the effect of inserting additional X or
  Z Pauli operators to the desired unitary U
  (depending on the position of the measured qubit).

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Accounting for randomness of measurement results

• Thus, we get, for example like
• P X Q Z R X
• when we want
• P Q R
• If we could somehow propagate through the Xs
  and Zs so that they stood at the left of P and
  Q, they could be corrected.
• But how does one do this for arbitrary U, which
  doesnt commute with X and Z?

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Clifford Group

• The Clifford group is the group of operators
  which normalise the Pauli group .
• which implies,

U preserved unchanged
Pauli may be changed by propagation
The Clifford group consists of CNOT, Hadamard,
Paulis and Pi/2 Phase gate (and products /
networks thereof). Only extra needed for
universal QC is qubit rotation.
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Rotation

• Rotations rotate a state in Bloch sphere, e.g.
• Propagation relations with Pauli operators
• Ux(-?),X0
• Ux(?) ZZ Ux(-?)
• i.e. sign of rotation is flipped, Pauli operator
  is unchanged.
• Does this mean that circuits with rotations
  cannot be realised in the QCC? No!

Ux(?)exp(-i??x /2)
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Rotation

• Ux(?)

If a minus eigenvalue is measured on the left
qubit, then we obtain an additional Pauli Z
operator, i.e.
Effective sign of rotation flipped!

• Compensate for this? Need to measure left qubit
  first!

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Arbitrary Rotation

• Euler decomposition Ux(?) Ux(?) Ux(?)

Arrows indicate the qubits whose measurement
outcome determines the sign of the basis. In
general, for the gates we proposed, effect
ofcorrecting extra Paulis only propagates
rightwards! Thus there are no contradictions in
temporal ordering!
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Clifford Group

• This still means that circuits involving solely
  the Clifford group (CNOT, Hadamard, Phase gate,
  Paulis) can be implemented in a single time
  step.
• c.f. In Gottesman-Knill theorem, such circuits
  are simulable by classical computers in
  polynomial time!
• Can remove Clifford part of an algorithm by
  tailoring specific graph states (see PRA).

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Example Network QFT

• The quantum network for QFT can be written

SWAP
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Example Network QFT

• Direct Implementation on QCC

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Example network Quantum Adder!
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Summary and References

• The working of the one-way quantum computer can
  well understood in terms of the formalism of
  stabilizers, which gives a formula for the design
  of gate measurement patterns.
• For more details and proofs
• R. Raussendorf, D.E. Browne and H. J. Briegel,
  Phys. Rev. A 68 022312 (2003).
• Good intro to stabilisers
• Nielsen and Chuang, Quantum Computation and
  Information p. 453