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Applications of randomized techniques in quantum information theory Debbie Leung, Caltech & U. Waterloo roll up our sleeves & prove a few things

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Unifying & Simplifying Measurement-based Quantum Computation Schemes Debbie Leung, Caltech & U. Waterloo Light, 95% math free may contain traces of physics

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Unifying & Simplifying Measurement-based Quantum Computation Schemes Debbie Leung, Caltech & U. Waterloo quant-ph/0404082,0404132 Joint work with Panos Aliferis, Andrew Childs, & Michael Nielsen Hashing ideas from Charles Bennett, Hans Briegel, Dan Browne, Isaac Chuang, Daniel Gottesman, Robert Raussendorf, Xinlan Zhou

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Universal QC schemes using only simple measurements:

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1WQC: Universal entangled initial state 1-qubit measurements Universal QC schemes using only simple measurements: 1) One-way Quantum Computer “1WQC” (Raussendorf & Briegel 00) : |+ i = |0 i +|1 i, : controlled-Z Cluster state: Can be easily prepared by (1) |+ i + controlled-Z, or ZZ, or (2) measurements of stabilizers e.g. X Z Z Z Z

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TQC: Any initial state (e.g. j 00 0 i ) 1&2-qubit measurements j 0 i ⋮ j 0 i u B ==== B ==== B ==== u B ==== u B ==== Universal QC schemes using only simple measurements: 2) Teleportation-based Quantum Computation “TQC” (Nielsen 01, L 01,03) Basic idea in each box: Bell XcZdUXcZdU U c,d

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1WQC: Universal entangled initial state 1-qubit measurements TQC: Any initial state (e.g. j 00 0 i ) 1&2-qubit measurements j 0 i ⋮ j 0 i u B ==== B ==== B ==== u B ==== u B ==== Universal QC schemes using only simple measurements: 1) One-way Quantum Computer “1WQC” (Raussendorf & Briegel 00) 2) Teleportation-based Quantum Computation “TQC” (Nielsen 01, L 01,03)

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Rest of talk: 0. Define simulation 1. Review 1-bit-teleportation Qn: are 1WQC & TQC related & can they be simplified? Here: derive simplified versions of both using “1-bit-teleportation” (Zhou, L, Chuang 00) (simplified version of Gottesman & Chuang 99) milk strawberry strawberry ice-cream & strawberry smoothy freeze & mix or mix & freeze 2. Derive intermediate simulation circuits (using much more than measurements) for a universal set of gates 3. Derive measurement-only schemes Ans: 1WQC = repeated use of the teleportation idea Then a big simplification suggests itself.

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Standard model for universal quantum computation : UU UU 0/1 UU U5U5 UnUn UU 0 : 0 : : time initial state Computation: gates from a universal gate set measure DiVincenzo 95 Wanted: a notion of “composable” elementwise-simulation

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Simulation of components up to known “leftist” Paulis (input to U), X a Z b (arbitrary known Pauli operator) (c,d) only depends on (a,b,k) k XaZbXaZb U XcZdXcZd (a,b) U U Intended evolution Simulation U simulates itself ,a,b UX a Z b = X c Z d U U Clifford group e.g. e.g. U X,Z: Pauli operators, a,b,c,d {0,1} U simulates I ,a,b UX a Z b = X c Z d U Pauli group e.g.

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UU UU 0/1 UU U5U5 UnUn 0 : 0 : Composing simulations to simulate any circuit : Simulation of circuit up to known “leftist” Paulis

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UU UU 0/1 U5U5 UnUn 0 : 0 Composing simulations to simulate any circuit : XaZbXaZb XaZbXaZb : UU Simulation of circuit up to known “leftist” Paulis XaZbXaZb UU UU State = (X a ) (X a )

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UU UU 0/1 U5U5 UnUn 0 : 0 Composing simulations to simulate any circuit : XaZbXaZb : UU XaZbXaZb XaZbXaZb Simulation of circuit up to known “leftist” Paulis State = (X a ) (X a )

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0/1 U5U5 UnUn 0 : 0 Composing simulations to simulate any circuit : XaZbXaZb UU UU XaZbXaZb : UU XaZbXaZb Simulation of circuit up to known “leftist” Paulis State = (X a ) (X a ) → X c Z d U 2 U 1

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UU UU 0/1 UU U5U5 UnUn 0 : 0 Composing simulations to simulate any circuit : XaZbXaZb : XcZdXcZd XcZdXcZd Simulation of circuit up to known “leftist” Paulis → X c Z d U 2 U 1 → X e Z f U 3 U 2 U 1 State = (X a ) (X a )

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UU UU 0/1 UU U5U5 UnUn 0 : 0 Composing simulations to simulate any circuit : : XaZbXaZb XaZbXaZb XaZbXaZb XaZbXaZb Simulation of circuit up to known “leftist” Paulis

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UU UU 0/1 UU U5U5 UnUn 0 : 0 Composing simulations to simulate any circuit : : XaZbXaZb XaZbXaZb XaZbXaZb XaZbXaZb Propagate errors without affecting the computation. Final measurement outcomes are flipped in a known (harmless) way. Simulation of circuit up to known “leftist” Paulis

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1-bit teleportation

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Z-Telepo (ZT) H c |0 i |i|i Zc|iZc|i H d c dc Teleportation without correction CNOT: Recall:Pauli’s: I, X, Z Hadamard: H H d X-rtation (XT) |0 i |i|i Xd|iXd|i NB. All simulate “I”.

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Simulating a universal set of gates: Z & X-rotations (1-qubit gates) & controlled-Z with mixed resources.

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Goal: perform Z rotation e iZ

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H c |0 i |i|i Zc|iZc|i Goal: perform Z rotation e iZ Z-Telep (ZT)

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H c |0 i |i|i Zc|iZc|i Goal: perform Z rotation e iZ c Z c e i(-1) a Z X a Z b | i XaZb|iXaZb|i H |0 i e i(-1) a Z = X a Z c+b e iZ | i Z-Telep (ZT) Input state = e i(-1) a Z X a Z b | i Xa eiZXa eiZ

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c Z-Telep (ZT) H c |0 i |i|i Zc|iZc|i XaZb|iXaZb|i H e i(-1) a Z Simulating a Z rotation e iZ X a Z c+b e iZ | i

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c Z-Telep (ZT) H H c d |0 i |i|i Zc|iZc|i X-Telep (XT) |0 i |i|i Xd|iXd|i X a Z c+b e iZ | i XaZb|iXaZb|i X a+d Z b e iX | i Simulating a Z rotation e iZ H |0 i e i(-1) a Z Simulating an X rotation e iX H d |0 i XaZb|iXaZb|i e i(-1) b X

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H d X-Telep (XT) |0 i |i|i Xd|iXd|i 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 -1 = C-Z: H d1 |0 i H d2 |0 i X a1 Z b1 X a2 Z b2 | i Simulating a C-Z X a1+d1 Z b1+a2+d2 X a2+d2 Z b2+a1+d1 C-Z | i

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From simulation with mixed resources to TQC -- QC by 1&2-qubit projective measurements only

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c X a Z c+b e iZ | i XaZb|iXaZb|i Simulating a Z rotation e iZ H |0 i e i(-1) a Z

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c X a+a2 Z c+b e iZ | i XaZb|iXaZb|i Simulating a Z rotation e iZ He i(-1) a Z An incomplete 2-qubit measurement, followed by a complete measurement on the 1st qubit. “X a2” |0 up to X a2 j HU V Z V†V† j U † XU V † ZV U † ZU k V†ZkVV†ZkV = A little fact: O = measurement of operator O

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c X a+a2 Z c+b e iZ | i XaZb|iXaZb|i Simulating a Z rotation e iZ “X a2”

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c X a+a2 Z c+b e iZ | i XaZb|iXaZb|i Simulating a Z rotation e iZ He i(-1) a Z “X a2” X a+d Z b+b2 e iX | i Simulating an X rotation e iX d XaZb|iXaZb|i e i(-1) b X “Z b2”

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c X a+a2 Z c+b e iZ | i XaZb|iXaZb|i Simulating a Z rotation e iZ He i(-1) a Z “X a2” X a+d Z b+b2 e iX | i Simulating an X rotation e iX d XaZb|iXaZb|i e i(-1) b X “Z b2” H d1 |0 i H d2 |0 i X a1 Z b1 X a2 Z b2 | i Simulating a C-Z X a1’ Z b1’ X a2’ Z b2’ C-Z | i

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c X a+a2 Z c+b e iZ | i XaZb|iXaZb|i Simulating a Z rotation e iZ He i(-1) a Z “X a2” X a+d Z b+b2 e iX | i Simulating an X rotation e iX d XaZb|iXaZb|i e i(-1) b X “Z b2” d1 d2 X a1 Z b1 X a2 Z b2 | i Simulating a C-Z H |0 i H X a1’ Z b1’ X a2’ Z b2’ C-Z | i

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c X a+a2 Z c+b e iZ | i XaZb|iXaZb|i Simulating a Z rotation e iZ He i(-1) a Z “X a2” X a+d Z b+b2 e iX | i Simulating an X rotation e iX d XaZb|iXaZb|i e i(-1) b X “Z b2” d1 d2 X a1 Z b1 X a2 Z b2 | i Simulating a C-Z X a1’ Z b1’ X a2’ Z b2’ C-Z | i Complete recipe for TQC based on 1-bit teleportation

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Aside: universality of 2-qubit meas is immediate! Bell c,d 2-qubit gate to be teleported 4-qubit state to be prepared d1 d2 Previous TQC with full teleportation: H |0 i H Simplified TQC with 1-bit teleportation: 2-qubit state to be prepared j HH Z j Z ZZ Z X k ZkZk =

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With slight improvements (see quant-ph/0404132): n-qubitm C-Z up to (m+1)n 1-qubit gates circuit Sufficient 2m 2-qubit meas 2m+n 1-qubit meas in TQC

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Deriving 1WQC-like schemes using gate simulations obtained from 1-bit teleportation 1WQC: Universal entangled initial state Feedforward 1-qubit measurement

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General circuit:... Alternating: (1) 1-qubit gates (2) nearest neighbor optional C-Z

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General circuit:... RzRz RxRx RzRz RzRz RxRx RzRz RzRz RxRx RzRz RzRz RxRx RzRz RzRz RxRx RzRz RzRz RxRx RzRz RzRz RzRz RzRz RzRz Alternating: (1) 1-qubit gates (2) nearest neighbor optional C-Z Z rotations + optional C-Z – X rotations – Z rotations + optional C-Z – X rotations –.... simulate these 2 things Euler-angle decomposition

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X a+d Z b e iX | i Simulating an X rotation e iX H d |0 i XaZb|iXaZb|i e i(-1) b X Adding an optional C-Z right before Z rotations c1 H |0 i c2 H |0 i e i(-1) a1 Z e i(-1) a2 Z X a1 Z b1 X a2 Z b2 | i X a1 Z b1+a2 k X a2 Z b2 +a1 k C-Z k | i X a1 Z c1+b1+a2 k X a2 Z c2+b2+a2 k e iZ e iZ C-Z k | i Will derive a method for optional C-Z later : the ability to choose to simulate I or C-Z

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|i|i optional c1 c2 H |0 i H e i(-1) a1 Z e i(-1) a2 Z H d1 |0 i e i(-1) b X H d2 |0 i e i(-1) b X c1’ c2’ H |0 i H e i(-1) a1 Z e i(-1) a2 Z H d1’ |0 i e i(-1) b X H d2’ |0 i e i(-1) b X C-Z+Z rotations –-- X rotations –-- C-Z+Z rotations –-- X rotations... Chaining up

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optional |i|i c1 c2 H |0 i H e i(-1) a1 Z e i(-1) a2 Z H d1 |0 i e i(-1) b X H d2 |0 i e i(-1) b X c1’ c2’ H |0 i H e i(-1) a1 Z e i(-1) a2 Z H d1’ |0 i e i(-1) b X H d2’ |0 i e i(-1) b X Use H|0 i = |+ i, HH=I H H H H H H H H H H H H H H H H = HH C-Z+Z rotations –-- X rotations –-- C-Z+Z rotations –-- X rotations... Chaining up

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|i|i c1 c2 H |+ i H e i(-1) a1 Z e i(-1) a2 Z c1 c2 H |+ i H e i(-1) a1 Z e i(-1) a2 Z d1 |+ i e i(-1) b X d2 |+ i e i(-1) b X H H optional d1 |+ i e i(-1) b X d2 |+ i e i(-1) b X H H C-Z+Z rotations –-- X rotations –-- C-Z+Z rotations –-- X rotations... Chaining up = |+ i Let, Then, initial state = |i|i

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Circuit dependent initial state: 3 qubits, 8 cycles of C-Z + 1-qubit rotations C-Z Z-rotations X-rotations

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H d1 |0 i H d2 |0 i X a1 Z b1 X a2 Z b2 | i Recall : simulating a C-Z Simulating an optional C-Z X a1’ Z b1’ X a2’ Z b2’ C-Z | i

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H d1 |0 i H d2 |0 i Recall : simulating a C-Z H d1 |0 i H d2 |0 i 1. Redrawing the 2nd input to the bottom: Simulating an optional C-Z

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2. Use symmetry: H d1 |0 i H d2 |0 i 1. Redrawing the 2nd input to the bottom: Just measures the parity of the 2 qubits It is equal to Simulating an optional C-Z j XjXj j

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2. Use symmetry: H |0 i H d2 d1 Simulating an optional C-Z Just measures the parity of the 2 qubits It is equal to j XjXj j

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H |0 i H d2 d1 Simulating an optional C-Z

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H |0 i H d2 d1 3. Use H = = HH Simulating an optional C-Z

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H |0 i H d2 d1 H H Simulating an optional C-Z 3. Use H = = HH

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H |0 i H H H H H = Simulating an optional C-Z H |0 i H d2 d1 H H

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3. Use H |0 i H H H H H = Simulating an optional C-Z H |0 i H d2 d1

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3. Use H|0 i =|+ i, |+ i d2 d1 “Remote C-Z” : Cousin of the remote CNOT by Gottesman98 H |0 i H d2 d1 Simulating an optional C-Z

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|+ i If one measures along {|0 i, |1 i }, the C-Zs labeled by ①② only acts like Z d1 Z d2 – simulating identity instead! d1 d2 ①②①② Simulating an optional C-Z |+ i d2 d1 “Remote C-Z” : Cousin of the remote CNOT by Gottesman98

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Simulating an optional C-Z, summary: |+ i d2 d1 |+ i d1 d2 simulates To do the C-Z:To skip the C-Z:

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Simulating an optional C-Z, summary: |+ i d2 d1 |+ i d1 d2 simulates To do the C-Z:To skip the C-Z: also simulates Do:Skip: |+ i Y Z up to Z-rotations

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Universal Initial state 3 qubits, 8 cycles

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Starting from the cluster state measure in Z basis

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Universal Initial state 3 qubits, 8 cycles

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Starting from the cluster state measure in Z basis

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Other universal initial state with other methods for optional C-Z

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Summary: Unified derivations, using 1-bit teleportation, for 1WQC & TQC + simplifications Details in quant-ph/0404082,0404132... Related results by Perdrix & Jorrand, Verstraete & Cirac but perhaps you don’t need to see them, you only need to remember what is a simulation (milk), what 1-bit teleportation does (strawberry), and the rest (mix/freeze) comes naturally.

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Summary: 1-bit teleportation has been used for systematic derivation of simplified constructions of fault tolerant gates. We have seen a similar use in deriving measurement-based QC. It leads to the remote C-Z/CNOT and programmable gate-array. Does it has a special role in quantum information theory?

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Open issues When it is already so simple ? Further optimizations?

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Open issues When it is already so simple ? Practically, the most interesting problems are likely to involve a mixture of resources, not just measurements. Measurement-based QC is most important as a conceptual tool. “Strawberry milkshake” taste much better with banana in it !

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Open issues mixture of resources, not just measurements. Running 1WQC in linear optics Nielsen 04, Drowne & Rudolph 04 Problem in linear optics: - C-Z difficult - C-Z probabilistically by teleportation trick Knill, Laflamme, Milburn01 Idea: Applying faulty C-Z to the data is expensive & failures are painful to repair. Instead, apply C-Z to build a cluster/graph state followed by 1-qubit measurements. Faulty C-Zs percolates the cluster state, but cheap to repair. Qn: optimize construction. Tradeoff between different methods to protect against percolations, e.g. good expensive C-Z vs redundant coding Qn: threshold under linear optics model? (1WQC: Raussendorf PhD thesis, Nielsen & Dawson 04)

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Open issues (what else to add to pure strawberry milkshake?) no threshold without fresh ancillas and interaction in 1WQC What are reasonable models for such resources ? What are reasonable models for the noise? Again, will be more experimentally motivated. e.g. Photons? Trapped ions? Quantum dots?

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