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Simplifying Measurement-based Quantum Computation Schemes

This paper explores the applications of randomized techniques in quantum information theory. It investigates simplified versions of measurement-based quantum computation schemes using "1-bit-teleportation" and analyzes the composable elementwise-simulation of circuits using known "leftist" Pauli operators.

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Simplifying Measurement-based Quantum Computation Schemes

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

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

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

  4. Universal QC schemes using only simple measurements:

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

  6. Basic idea in each box: j U c,d Bell j0i ⋮ j0i XcZdUj = = B = = = = u B B = = = = u u B B Universal QC schemes using only simple measurements: 2) Teleportation-based Quantum Computation “TQC” (Nielsen 01, L 01,03) TQC: • Any initial state (e.g. j00L0i) • 1&2-qubit measurements

  7. 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 01,03) 1WQC: • Universal entangled initial state • 1-qubit measurements TQC: • Any initial state (e.g. j00L0i) • 1&2-qubit measurements

  8. 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) Ans: 1WQC = repeated use of the teleportation idea Then a big simplification suggests itself. 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

  9. 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 Wanted: a notion of “composable” elementwise-simulation

  10. 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} Intended evolution Simulation j U j U (c,d) only depends on (a,b,k) U simulates I  j,a,b UXaZbj = XcZdj  U  Pauli group e.g.

  11. 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

  12. 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 State = (Xa 0)  (Xa 0)  

  13. 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 State = (Xa 0)  (Xa 0)  

  14. 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 State = (Xa 0)  (Xa 0)   →XcZdU2U10  0  

  15. 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 State = (Xa 0)  (Xa 0)   →XcZdU2U10  0   →Xe ZfU3U2U10  0  

  16. 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

  17. 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.

  18. 1-bit teleportation

  19. 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

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

  21. Goal: perform Z rotation eiqZ

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

  23. XaZb|i ei(-1)aqZ H c |0i Zcei(-1)aqZ XaZb|i = Xa Zc+beiqZ|i Z-Telep (ZT) Goal: perform Z rotation eiqZ |i H c |0i Zc|i XaeiqZ Input state = ei(-1)aqZ XaZb|i

  24. 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

  25. 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

  26. 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: =

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

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

  29. U H j V† V = j U†ZU U†XU V†ZV k V†ZkV O = measurement of operator O Z 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 . A little fact:

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

  31. 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”

  32. Xa1’ Zb1’  Xa2’ Zb2’ C-Z|i 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” Xa1Zb1 Xa2Zb2|i Simulating a C-Z d1 d2 |0i H |0i H

  33. Xa1’ Zb1’  Xa2’ Zb2’ C-Z|i 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” Xa1Zb1 Xa2Zb2|i Simulating a C-Z d1 d2 |0i H |0i H

  34. Xa1’ Zb1’  Xa2’ Zb2’ C-Z|i Simulating a Z rotation eiqZ XaZb|i ei(-1)aqZ H c Xa+a2 Zc+beiqZ|i “Xa2” Complete recipe for TQC based on 1-bit teleportation Simulating an X rotation eiqX XaZb|i ei(-1)bqX d Xa+d Zb+b2eiqX|i “Zb2” Xa1Zb1 Xa2Zb2|i Simulating a C-Z d1 d2

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

  36. With slight improvements (see quant-ph/0404132): n-qubitm C-Z up to (m+1)n1-qubit gatescircuitSufficient 2m 2-qubit meas 2m+n 1-qubit meas in TQC

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

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

  39. 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

  40. Xa1 Zb1+a2k Xa2 Zb2 +a1k C-Zk|i Xa1 Zc1+b1+a2k Xa2 Zc2+b2+a2k eiq1Zeiq2ZC-Zk|i Simulating an X rotation eiqX XaZb|i ei(-1)bqX d |0i Xa+d ZbeiqX|i H Adding an optional C-Z right before Z rotations ei(-1)a1q1Z H c1 Xa1Zb1 Xa2Zb2|i ei(-1)a2q2Z c2 H |0i |0i Will derive a method for optional C-Z later : the ability to choose to simulate I or C-Z

  41. 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

  42. 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

  43. 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

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

  45. Simulating an optional C-Z Recall : simulating a C-Z Xa1Zb1 Xa2Zb2|i d1 d2 |0i H Xa1’ Zb1’  Xa2’ Zb2’ C-Z|i |0i H

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

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

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

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

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

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