Unifying Simplifying Measurementbased Quantum Computation Schemes Debbie Leung

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

Universal QC schemes using only simple measurements:

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

strawberry ice-cream & strawberry smoothy Qn: are 1 WQC & 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

Standard model for universal quantum computa Di. Vincenzo 95 : U 5 0 U 3 U 1 U 4 0 : : Un U : 2 0 initial state Computation: gates from a universal gate set 0/1 0/1 time measure

Simulation of components up to known “leftist” Pau e. g. U j (input to U), Xa. Zb (arbitrary known Pauli operator) X, Z: Pauli operators, a, b, c, d {0, 1} Wanted j Simulation U U j e. g. U simulates itself j , a, b UXa. Zb j = Xc. Zd. U j U Clifford group Xa. Zb j k (a, b) Xc. Zd U j (c, d) only depends on (a, b, k)

Simulation of circuit up to known “leftist” Paulis Composing simulations to simulate any circuit : 0 0 : : 0 U 3 U 1 U 2 : 0/1 U 5 Un 0/1

Simulation of circuit up to known “leftist” Paulis Composing simulations to simulate any circuit : 0 X Z : : 0 X Z a a a b b b U 1 U 2 0/1 U 5 U 3 : Un 0/1

Simulation of circuit up to known “leftist” Paulis Composing simulations to simulate any circuit : 0 X Z U 1 : : 0 X Z U 2 a b a b 0/1 U 5 U 3 : Un 0/1

Simulation of circuit up to known “leftist” Paulis Composing simulations to simulate any circuit : 0 0 : : 0 Xa Zb U 1 Xa Zb U 2 Xa Zb 0/1 U 5 U 3 : Un 0/1

Simulation of circuit up to known “leftist” Paulis Composing simulations to simulate any circuit : 0 0 : : 0 Xc Zd U 3 U 1 U 2 Xa Zb 0/1 U 5 Xc Zd : Un 0/1

Simulation of circuit up to known “leftist” Paulis Composing simulations to simulate any circuit : 0 0 : : 0 U 1 U 3 U 5 Xa Zb 0/1 Xa Zb U 2 : Un X a Z b 0/1

Simulation of circuit up to known “leftist” Paulis Composing simulations to simulate any circuit : 0 0 : : 0 U 1 U 5 U 3 Xa Zb 0/1 Xa Zb U 2 : Un X a Z b 0/1 Propagate errors without affecting the computation. Final measurement outcomes are flipped in a known (harmless) way.

1 -bit teleportation

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

Goal: perform Z rotation eiq. Z

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

X-Telep (XT) Simulating a C-Z | i Xa 1 Zb 1 Xa 2 Zb 2| i |0 i d H d 1 d 2 Xd| i |0 i H H Xa 1+d 1 Zb 1+a 2+d 2 Xa 2+d 2 Zb 2+a 1+d 1 C-Z | i C-Z: 1 = 0 0 0 0 1 0 0 -1

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

Simulating a Z rotation eiq. Z Xa. Zb| i ei(-1)aq. Z |0 i H c Xa Zc+b eiq. Z| i

Simulating a Z rotation eiq. Z Xa. Zb| i ei(-1)aq. Z “Xa 2” |0 up to Xa 2 H c Xa+a 2 Zc+b eiq. Z| i An incomplete 2 -qubit measurement, followed by a complete measurement on the 1 st qubit.

Simulating a Z rotation eiq. Z Xa. Zb| i “Xa 2” c Xa+a 2 Zc+b eiq. Z| i

Simulating a Z rotation eiq. Z Xa. Zb| i ei(-1)aq. Z c H Xa+a 2 Zc+b eiq. Z| i “Xa 2” Simulating an X rotation eiq. X b Xa. Zb| i ei(-1) q. X “Zb 2” d Xa+d Zb+b 2 eiq. X| i

Simulating a Z rotation eiq. Z Xa. Zb| i ei(-1)aq. Z c H Xa+a 2 Zc+b eiq. Z| i “Xa 2” Simulating an X rotation b Xa. Zb| i ei(-1) q. X “Zb 2” Simulating a C-Z Complete recipe for TQC based on 1 -bit teleportation eiq. X d Xa+d Zb+b 2 eiq. X| i Xa 1 Zb 1 Xa 2 Zb 2| i d 1 d 2 Xa 1’ Zb 1’ Xa 2’ Zb 2’ C-Z | i

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

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

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

General circuit: Rz Rx Rz Rz Rz . . . Rz Rx Rz Alternating: (1) 1 -qubit gates (2) nearest neighbor optional C-Z Euler-angle decomposition 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. . . optional | i ei(-1)a 1 q 1 Z H ei(-1)a 2 q 2 Z H |0 i |0 i |0 i H c 1 c 2 ei(-1)bq. X d 1 ei(-1)bq. X d 2

Chaining up C-Z+Z rotations –-- X rotations –-- C-Z+Z rotations –-- X rotations. . . optional | i ei(-1)a 1 q 1 Z H ei(-1)a 2 q 2 Z H |0 i H Hei(-1)bq. X H i(-1)bq. X e Use c 1 c 2 H H H|0 i = |+i, HH=I d 1 d 2 ei(-1)a 1 q 1 Z H ei(-1)a 2 q 2 Z H |0 i H = H H |0 i H c 1 c 2 b Hei(-1) q. X d 1 Hei(-1)bq. X d 2

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

Simulating an optional C-Z: To do the C-Z: d 1 d 2 |+i turns out equiv to Gottesman’s remote-CNOT Simulating a C-Z d 1 d 2 |0 i H H

Simulating an optional C-Z: To do the C-Z: |+i Thus: To skip the C-Z: d 1 d 2 |+i d 1 d 2 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 1 WQC & TQC + simplifications Details 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.