Resources for Measurement Based Quantum Computation Quantum CarryLookahead
Resources for Measurement. Based Quantum Computation Quantum Carry-Lookahead Adder Presented in October 21 st, 2008 Agung Trisetyarso Keio University
Abstract • Presenting the design of quantum carry-lookahead adder using measurement-based quantum computation • QCLA utilizes MBQC`s ability to transfer quantum states in unit time to accelerate addition • QCLA 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
Content • • Introduction of Raussendorf Theorem in Cluster State Introduction of Quantum Carry-Lookahead Adder – In-place circuit – Out-of-place circuit • Quantum Adders: 1. Quantum Ripple-Carry Adders 2. 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
Raussendorf Theorem in Cluster State • Quantum computation in simplest (abstract) in cluster state system – Clifford Group – Linear Transformation – No Teleportation – No Measurement. Driven – Teleportation – Measurement-Driven
Quantum computation in simplest (abstract) system • IDENTITY Gate • NOT Gate
Questions: • How to deliver quantum information in real physical systems?
Raussendorf, Briegel and Browne’s Theorem AQIS`08, KIAS
Raussendorf, Briegel and Browne’s Theorem • Initial Eigenvalue Equations • Measurement • Final Eigenvalue Equations
Properties of MBQC
Quantum computation in cluster state • IDENTITY Gate • NOT Gate
Quantum computation in cluster state
Measurement Step • Three qubits on Machine Cluster (CM) are measured in one time.
Measurement Step • Three qubits on Machine Cluster (CM) are measured in one time.
Measurement Step • 65 qubits on Machine Cluster (CM) are measured in first time. • 7 qubits on CM are measured on second time
Quantum Adder Ripple Carry Adder Carry-Lookahead Adder
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
Vedral, Barenco and Ekert Adder Circuit = = =
The Carry-Lookahead Adder • Using generating and propagating carries concepts • The addition of two 1 -digit inputs ai and bi is said to generate if the addition will always carry, regardless of whethere is an input carry • The addition of two 1 -digit inputs ai and bi is said to propagate if the addition will carry whenever there is an input carry
Implementation of CLA into Quantum Circuit Quantum Carry-Lookahead Adder Addition Circuit Out-of-place (5 procedures) Carry Computation Circuit (3 procedures) In-place (10 procedures)
Carry Computation Circuit • Procedures to determine the rounds: Where: n = logical qubits t = sequences of rounds m = number of rounds
Out-of-place Quantum Carry. Lookahead Adder • The circuit aims to perform by the following procedures:
Out-of-place QCLA circuit • • Red : G-Rounds Blue: P-Rounds Green: C-Rounds Black: SUMBlocks
Performances and Requirements of Out-of-place MBQC QCLA
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:
In-place Quantum Carry. Lookahead Adder
In-place QCLA circuit • • Red : G-Rounds Blue: P-Rounds Green: C-Rounds Black: SUMBlocks
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.
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
“Bend a Network”Inplace QCLA Circuit • Reduce the horizontal resources, but spent more vertically
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 )
Performances and Requirements of Out-of-place MBQC QCLA Circuit
Total Resources Qubits Example for n=10 => Total Qubits or in-place circuit = 14657
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 )
Size Comparison of Out-of-place, In -place MBQC and MBQC VBE
Depth Comparison of Out-of-place, In-place MBQC and MBQC VBE
Optimized-in-place MBQC QCLA circuit • Diamond-like form circuit, spatial resources optimation ≈ 26 % from in-place MBQC QCLA circuit.
Optimation of MBQC QCLA Circuit Or, ≈ Example for n=10 => Removed Qubits/Total Qubits = 3822/14657 ≈ 26 %
Conclusion • The resources to perform quantum carrylookahead 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
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)
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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