Resources for Measurement-Based Quantum Computation Quantum Carry-Lookahead Adder

1. Resources for Measurement-Based Quantum Computation Quantum Carry-Lookahead Adder Presented in October 21st, 2008 Agung Trisetyarso Keio University

2. Abstract • Presenting the design of quantum carry-lookahead adder using measurement-based quantum computationQCLA utilizes MBQC`s ability to transfer quantum states in unit time to accelerate additionQCLA is faster than a quantum ripple-carry adder; QCLA has logarithmic depth while ripple adders have linear depth QCLA requires a cluster state that is an order of magnitude larger“Bend a Network” method results ≈26 % spatial resources optimation for in-place MBQC QCLA circuit

3. Content • Introduction of Raussendorf Theorem in Cluster State • Introduction of Quantum Carry-Lookahead Adder • In-place circuit • Out-of-place circuit • Quantum Adders: • Quantum Ripple-Carry Adders • Quantum Carry-Lookahead Adder • Performance of Measurement-Based Quantum Carry-Lookahead Adder Circuit • Conclusions • Next Research Proposal • Spin Cluster Qubits in All-Silicon Quantum Computer • Resources for Silica-on-Silicon Waveguide MBQC QCLA with Photon • MBQC Circuit with Fault-Tolerant

4. Quantum computation in simplest (abstract) system Linear Transformation No Teleportation No Measurement-Driven Quantum computation in cluster state Clifford Group Teleportation Measurement-Driven Raussendorf Theorem in Cluster State

5. IDENTITY Gate NOT Gate Quantum computation in simplest (abstract) system

6. Questions: • How to deliver quantum information in real physical systems?

7. Raussendorf, Briegel and Browne’s Theorem AQIS`08, KIAS

8. Raussendorf, Briegel and Browne’s Theorem • Initial Eigenvalue Equations • Measurement • Final Eigenvalue Equations

9. Properties of MBQC

10. IDENTITY Gate NOT Gate Quantum computation in cluster state

11. Quantum computation in cluster state

12. Measurement Step • Three qubits on Machine Cluster (CM) are measured in one time.

13. Measurement Step • Three qubits on Machine Cluster (CM) are measured in one time.

14. Measurement Step • 65 qubits on Machine Cluster (CM) are measured in first time. • 7 qubits on CM are measured on second time

15. Quantum Adder

16. Ripple Carry Adder • Multiplying full adders used with the carry ins and carry outs chained • The correct value of the carry bit ripples from one bit to the next. • The Depth is O(N) or Polynomial -> relatively slow, since each full adder must wait for the carry bit to be Calculated from the previous full adder

17. Vedral, Barenco and Ekert Adder Circuit = = =

18. The Carry-Lookahead Adder • Using generating and propagating carries concepts • The addition of two 1-digit inputs aiand bi is said to generate if the addition will always carry, regardless of whether there is an input carry • The addition of two 1-digit inputs aiand biis said to propagate if the addition will carry whenever there is an input carry

19. Implementation of CLA into Quantum Circuit

20. Carry Computation Circuit • Procedures to determine the rounds: Where: n = logical qubits t = sequences of rounds m = number of rounds

21. Out-of-place Quantum Carry-Lookahead Adder • The circuit aims to perform by the following procedures:

22. Out-of-place QCLA circuit • Red : G-Rounds • Blue: P-Rounds • Green: C-Rounds • Black: SUM-Blocks

25. Performances and Requirements of Out-of-place MBQC QCLA

26. In-place Quantum Carry-Lookahead Adder • The in-place circuit aims to erasure every unnecessary subregisters output. • The additional circuit is that it should perform: • The implementation in Quantum circuit is expressed in following procedures:

27. In-place Quantum Carry-Lookahead Adder

28. In-place QCLA circuit • Red : G-Rounds • Blue: P-Rounds • Green: C-Rounds • Black: SUM-Blocks

29. Bend a Network Circuit • One may imagine the logical qubits as • traveling through pipes on a two-dimensional surface. • Horizontal and vertical axes both represent spatial axes, not temporal.

30. MBQC Form of VBE Robert Raussendorf, Daniel E. Browne, and Hans J. Briegel, Phys. Rev. A 68, 022312 (2003) • Optimized for space, but still linear depth • spatial resources = 304 n

31. “Bend a Network”In-place QCLA Circuit • Reduce the horizontal resources, but spent more vertically

32. Out-of-place MBQC QCLA circuit • For n=10, consist of: • 4 addition circuits • 9 carry networks (2 Propagate, 3 Generate, 2 Inverse Propagate and 2 Carry networks )

33. Performances and Requirements of Out-of-place MBQC QCLA Circuit

34. Total Resources Qubits Example for n=10 => Total Qubits or in-place circuit = 14657

35. In-place MBQC QCLA circuit • For n=10, consist of: • 8 addition circuits • 18 carry networks (4 Propagate, 6 Generate, 4 Inverse Propagate and 4 Carry networks )

36. Size Comparison of Out-of-place, In-place MBQC and MBQC VBE

37. Depth Comparison of Out-of-place, In-place MBQC and MBQC VBE

38. Optimized-in-place MBQC QCLA circuit • Diamond-like form circuit, spatial resources optimation ≈ 26 % from in-place MBQC QCLA circuit.

39. Optimation of MBQC QCLA Circuit Or, ≈ Example for n=10 => Removed Qubits/Total Qubits = 3822/14657 ≈ 26 %

40. Conclusion • The resources to perform quantum carry-lookahead adder in cluster state = f(logical qubits, width and number of qubits in quantum gates) • “Bend a Network” changes Manhattan grid form to Diamond-like form in MBQC QCLA circuit. • Optimation ≈ 26 % spatial resources

41. Future Works(1): • “Resources for Photonic Cluster State Computation Quantum Carry-Lookahead Adder Circuit” References: 1. Devitt et al., Topological Cluster State with Photons, quant-ph ... 2. Stephens et. al, Deterministic optical quantum computer using photonic modules, quant-ph ... 3. Politi et. al, Silica-on-Silicon Waveguide Quantum Circuits, Science 320, 646 (2008)

42. Future Works (2): • “Resources for Quantum-dot cluster state computing Quantum Carry-Lookahead Adder” References: 1. Weinstein et. al, Quantum-dot cluster-state computing with encoded qubits, PRA 72, 020304(R) (2005) 2. Meier et. al, Quantum Computing with Spin Cluster Qubits, PRL (2003) 3. Meier et. al, Quantum Computing with antiferromagnetic spin clusters, PRB 68, 134417 (2003) 4. Skinner et. al, Hydrogenic Spin Quantum Computing in Silicon: A Digital Approach, PRL 2003 5. J.Levy, Universal Quantum Computation with Spin-1/2 Pairs and Heissenberg Exchange, PRL 2002 6. Rahman et. al, High Precision Quantum Control of Single Donor Spins in Silicon, PRL 2007.

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