Universal QC schemes using only simple measurements:

Unifying & Simplifying Measurement-based Quantum Computation Schemes Debbie Leung, Caltechquant-ph/0404082,0404132Joint works with Panos Aliferis, Andrew Childs, & Michael NielsenHashing ideas from Charles Bennett, Hans Briegel, Dan Browne, Isaac Chuang, Daniel Gottesman, Robert Raussendorf, Xinlan Zhou

Universal QC schemes using only simple measurements:

j0i ⋮ j0i = = B = = = = u B B = = = = u u B 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) 1WQC: • Universal entangled initial state • 1-qubit measurements TQC: • Any initial state (e.g. j00L0i) • 1&2-qubit measurements

strawberry ice-cream & strawberry smoothy 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) Rest of talk: 0. Define simulation 1. Review 1-bit-teleportation milk strawberry 2. Derive intermediate simulation circuits (using much more than measurements) for a universal set of gates 3. Derive measurement-only schemes freeze & mix or mix & freeze

00 : : 0 0/1 U5 U3 U1 0/1 : U4 : Un U2 0/1 time Standard model for universal quantum computation : DiVincenzo 95 Computation: gates from a universal gate set initial state measure

j XaZb k j XcZd U (a,b) e.g. U simulates itself  j,a,b UXaZbj = XcZdUj  U  Clifford group Simulation of componentsup to known “leftist” Paulis e.g. U  j (input to U),  XaZb (arbitrary known Pauli operator) X,Z: Pauli operators, a,b,c,d {0,1} Wanted Simulation j U j U (c,d) only depends on (a,b,k)

Simulation of circuitup to known “leftist” Paulis Composing simulations to simulate any circuit : 00 : : 0 0/1 U5 U3 U1 0/1 : Un U2 0/1

Simulation of circuitup to known “leftist” Paulis Composing simulations to simulate any circuit : 00 : : 0 XaZb 0/1 U5 U3 U1 U1 XaZb 0/1 : Un XaZb U2 U2 0/1

Simulation of circuitup to known “leftist” Paulis Composing simulations to simulate any circuit : 00 : : 0 XaZb 0/1 U5 U3 U1 XaZb 0/1 : Un XaZb U2 0/1

U1 U2 Simulation of circuitup to known “leftist” Paulis Composing simulations to simulate any circuit : 00 : : 0 XaZb 0/1 U5 U3 XaZb 0/1 : Un XaZb 0/1

Simulation of circuitup to known “leftist” Paulis Composing simulations to simulate any circuit : 00 : : 0 XcZd 0/1 U5 U3 U1 XcZd 0/1 : Un XaZb U2 0/1

Simulation of circuitup to known “leftist” Paulis Composing simulations to simulate any circuit : 00 : : 0 XaZb 0/1 U5 U3 U1 XaZb 0/1 XaZb : Un XaZb U2 0/1

Simulation of circuitup to known “leftist” Paulis Composing simulations to simulate any circuit : 00 : : 0 XaZb 0/1 U5 U3 XaZb U1 0/1 XaZb : Un XaZb U2 0/1 Propagate errors without affecting the computation. Final measurement outcomes are flipped in a known (harmless) way.

1-bit teleportation

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

Simulating 1-qubit gates & controlled-Z with mixed resources.

Goal: perform Z rotation eiqZ

Z-Telep (ZT) Goal: perform Z rotation eiqZ |i H c |0i Zc|i

Z-Telep (ZT) Simulating a Z rotation eiqZ XaZb|i |i ei(-1)aqZ H c H c |0i Xa Zc+beiqZ|i |0i Zc|i

Z-Telep (ZT) Simulating a Z rotation eiqZ XaZb|i |i ei(-1)aqZ H c H c |0i Xa Zc+beiqZ|i |0i Zc|i X-Telep (XT) Simulating an X rotation eiqX XaZb|i ei(-1)bqX |i d d |0i Xa+d ZbeiqX|i |0i H H Xd|i

Simulating a C-Z Xa1Zb1 Xa2Zb2|i d1 d2 |0i H |0i H Xa1+d1Zb1+a2+d2Xa2+d2Zb2+a1+d1 C-Z|i X-Telep (XT) |i d |0i H Xd|i 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 -1 C-Z: =

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

Simulating a Z rotation eiqZ XaZb|i ei(-1)aqZ H c |0i Xa Zc+beiqZ|i

Simulating a Z rotation eiqZ XaZb|i ei(-1)aqZ H c Xa+a2 Zc+beiqZ|i “Xa2” |0 up to Xa2 An incomplete 2-qubit measurement, followed by a complete measurement on the 1st qubit .

Simulating a Z rotation eiqZ XaZb|i c Xa+a2 Zc+beiqZ|i “Xa2”

Simulating a Z rotation eiqZ XaZb|i ei(-1)aqZ H c Xa+a2 Zc+beiqZ|i “Xa2” Simulating an X rotation eiqX XaZb|i ei(-1)bqX d Xa+d Zb+b2eiqX|i “Zb2”

See more improvements in quant-ph/0404132Punchline : 2m 2-qubit & 2m+n 1-qubit measurements for a circuit of n qubits with m C-Zs & arbitrary 1-qubit gates

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

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

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

Chaining up C-Z+Z rotations –-- X rotations –-- C-Z+Z rotations –-- X rotations ... c1 c1 c2 c2 ei(-1)bqX d1 ei(-1)bqX d2 |0i H |0i H optional ei(-1)a1q1Z H |i ei(-1)a2q2Z H |0i |0i ei(-1)a1q1Z H ei(-1)a2q2Z H |0i |0i ei(-1)bqX d1 ei(-1)bqX d2 |0i H |0i H

Chaining up C-Z+Z rotations –-- X rotations –-- C-Z+Z rotations –-- X rotations ... = c1 c1 H H c2 c2 optional Use H|0i = |+i, HH=I ei(-1)a1q1Z H |i ei(-1)a2q2Z H |0i H H |0i H H ei(-1)bqX H H d1 H H ei(-1)bqX d2 |0i H |0i ei(-1)a1q1Z H H ei(-1)a2q2Z H |0i H H |0i H H ei(-1)bqX d1 H H H ei(-1)bqX H d2 |0i H |0i H

Chaining up C-Z+Z rotations –-- X rotations –-- C-Z+Z rotations –-- X rotations ... c1 c1 c2 c2 |i = |+i Let , optional Then, initial state = ei(-1)a1q1Z H |i ei(-1)a2q2Z H |+i |+i ei(-1)bqX H d1 H ei(-1)bqX d2 |+i |+i ei(-1)a1q1Z H ei(-1)a2q2Z H |+i |+i ei(-1)bqX d1 H ei(-1)bqX H d2 |+i |+i

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

|+i d1  |+i Simulating a C-Z d2   d1  d2 |0i H |0i H Simulating an optional C-Z: To do the C-Z: turns out equiv to Gottesman’s remote-CNOT

|+i d1 |+i d2 |+i d1 |+i d2 Simulating an optional C-Z: To do the C-Z: To skip the C-Z: Thus: gives the ability for simulating

Universal Initial state 3 qubits, 8 cycles

Starting from the cluster state measure in Z basis

Universal Initial state 3 qubits, 8 cycles

Starting from the cluster state measure in Z basis

Summary: Unified derivations, using 1-bit teleportation, for 1WQC & TQC + simplificationsDetails in quant-ph/0404082,0404132 ... 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. Related results by Perdrix & Jorrand, Cirac & Verstraete.

Universal QC schemes using only simple measurements:

Universal QC schemes using only simple measurements:

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