Adiabatic Quantum Transistors Dave Bacon, Gregory Crosswhite University of Washington & Steve Flammia Perimeter Institute Workshop on Quantum Algorithms, Computational Models and Foundations of Quantum Mechanics UBC, Vancouver, 23 July 2010 A zoo of quantum computational models Measurement-based Topological Circuit model Adiabatic Holonomic A zoo of quantum computational models Which one (if any!) Measurement-based Topological will lead to an actual quantum computer? CircuitThis model talk: try to combine Adiabatic aspects of all of these models to devise a new architecture for Holonomic quantum computing A review of the zoo Adiabatic evolution offers robustness to timing and control errors that exist in the circuit model Measurement-based ErrorsTopological are suppressed by the spectral gap It is unknown if it is fault tolerant (without additional assumptions) and lack of modularity makes Circuitit difficult model to analyze theoretically Adiabatic Holonomic A review of the zoo Holonomic QC is also robust to timing errors, and some (fewer) types of control errors Measurement-based CanTopological be made fault tolerant Oreshkov Brun Lidar, PRL 2009 Typically requires simultaneous control of multiple parameters to achieve non-trivial Circuit modelgeometric phases Adiabatic Holonomic A review of the zoo Topological quantum phases are insensitive to local perturbations Measurement-based Topological Bravyi Hastings Michalakis 2010 Naturally long-lived quantum memory Circuit model Adiabatic Sensitive to finite temperature, and still requires active error correction. Also,Holonomic initialization is difficult. A review of the zoo Measurement-based Topological Very minimal requirements: only local measurements, which every scheme uses anyway Simple initial statesCircuit (relatively speaking) can be used as model the entangled resources. Holonomic There isAdiabatic absolutely nothing disadvantageous about measurement-based QC A review of the zoo Circuit model provides the mostMeasurement-based natural language for programming quantum computers and designing Topological quantum algorithms Circuit model Direct implementation involves pulsed gates and a huge Holonomic amountAdiabatic of control… very challenging, to say the least. Adiabatic teleportation single qubit A |ψ B C Bell pair |Φ This is a ground state of Hi = −X2 X3 − Z2 Z3 (could also use the exchange interaction) Bacon STF, PRL 2009 Related: Oreshkov Brun Lidar, PRL 2009; Oreshkov, PRL 2009 Adiabatic teleportation Bell pair A B |Φ C |ψ single qubit This is a ground state of Hf = −X1 X2 − Z1 Z2 Adiabatic teleportation A |ψ B C H(t) = (1 − t)Hi + tHf Adiabatic teleportation A B C H(t) = (1 − t)Hi + tHf Adiabatic teleportation A B C H(t) = (1 − t)Hi + tHf T exp −i T 0 dτ H(τ ) Adiabatic teleportation A B C H(t) = (1 − t)Hi + tHf Adiabatic teleportation A B C H(t) = (1 − t)Hi + tHf Notice that the ground space is stabilized by XXX and ZZZ for all t. Adiabatic teleportation A B C |ψ H(t) = (1 − t)Hi + tHf Notice that the ground space is stabilized by XXX and ZZZ for all t. Adiabatic teleportation A B C |ψ The adiabatic evolution acts like a post-selected teleportation! Notice that the ground space is stabilized by XXX and ZZZ for all t. Adiabatic gate teleportation A |ψ † U3 H(t)U3 B C = (1 − † t)U3 Hi U3 Gottesman Chuang 1999 + tHf Adiabatic gate teleportation A † U3 H(t)U3 B C = (1 − † t)U3 Hi U3 Gottesman Chuang 1999 + tHf Adiabatic gate teleportation A B † U3 H(t)U3 = (1 − C U3 |ψ † t)U3 Hi U3 Gottesman Chuang 1999 + tHf Adiabatic gate teleportation A B † U3 H(t)U3 = (1 − C U3 |ψ † t)U3 Hi U3 + tHf Now the adiabatic evolution teleports the unitary onto the qubit. Gottesman Chuang 1999 Universality A B C Ua |ψ Universality A Ub Ua |ψ B C Universality A B C Ub Ua |ψ Etc... but what about two qubit gates? Universality A B C A’ B’ C’ Universality A B C A’ B’ C’ Universality A B C A’ B’ C’ Two-qubit gates introduce 3-body terms... Universality A B C A’ B’ C’ Two-qubit gates introduce 3-body terms... Universality A B C A’ B’ C’ Two-qubit gates introduce 3-body terms... to get rid of them, use perturbation gadgets. Universal, 2-body A B C D A’ B’ C’ D’ Z Z coupling Our perturbation gadgets: Bartlett & Rudolph 2006 Universal, 2-body A B C D A’ B’ C’ D’ Z Z coupling Our perturbation gadgets: Bartlett & Rudolph 2006 Universal, 2-body A B C D Qubit encoded in subspace |00 , |11 A’ B’ C’ D’ Z Z coupling Our perturbation gadgets: Bartlett & Rudolph 2006 Universal, 2-body A B C D A’ B’ C’ D’ Z Z coupling Our perturbation gadgets: Bartlett & Rudolph 2006 Universal, 2-body A B C D Now qubits are encoded locally A’ B’ C’ D’ Z Z coupling Our perturbation gadgets: Bartlett & Rudolph 2006 Universal, 2-body A A’ B B’ C C’ D D’ Gate fidelity = 1 − Θ λ 2 Gap = Θ(λ) Ratio of energy scales = λ = Our perturbation gadgets: Bartlett & Rudolph 2006 1-d architecture A A A A A A A A A B B B B B B B B B C C C C C C C C C D D D D D D D D D 1 2 3 ... n Adiabatic Code Deformation Energy Quantum errorcorrecting codespace Quantum errorcorrecting codespace Time Must be degenerate throughout the entire evolution; any splittings are errors that need to be coded for and corrected. Bombin Delgado, J. Phys. A 2009 “Open-loop” holonomy, Kult Aberg Sjoqvist, PRA 2006 Why is this interesting? Adiabatic holonomic evolution offers robustness to timing and control errors that exist in the circuit model Excitations are suppressed by the constant gap “Ground state” errors can be corrected via coding It is modular, and hence as easy to program as the circuit model Uses only control between subsystems, not levels Gates are prepared offline, leading to fewer errors It leads to more results of interest to theorists... One Way QC |0 Rθ • |0 H FE FE Create Entangled State Raussendorf Briegel, PRL 2001 One Way QC |0 Rθ • |0 H FE FE X X Adaptively measure to enact circuit Raussendorf Briegel, PRL 2001 One Way QC |0 Rθ • |0 H X X X±! X X Y Y Y FE FE Adaptively measure to enact circuit Raussendorf Briegel, PRL 2001 One Way QC X X X±! |0 Rθ • |0 H Y Y X FE FE Y Y Y X X Y X Y Y Y X X Y Adaptively measure to enact circuit Raussendorf Briegel, PRL 2001 One Way QC X X X±! |0 Rθ • |0 H Y Y X FE FE Y Y Y Z X X Z Y X Y Y Y X X Y Adaptively measure to enact circuit Raussendorf Briegel, PRL 2001 Cluster State Hamiltonian Z Z X Z Z Sv = X v Zw w adjacent to v Cluster state is ground state of HC = −∆ Sv v Again, it’s possible to use gadgets to make only 2-qubit interactions Bartlett Rudolph, PRA 2006 Adiabatic One-way QC 1 2 3 ... n-1 n Sj = Zj−1 Xj Zj+1 n−1 H = −Zn−1 Xn − Sj j=2 Suppose we prepare |+ on the first physical qubit Turn on -X fields and turn off cluster state coupling Bacon Flammia, arXiv:0912.2098 Adiabatic One-way QC 1 2 3 ... -X n-1 n Sj = Zj−1 Xj Zj+1 n−1 H = −Zn−1 Xn − Sj j=2 Suppose we prepare |+ on the first physical qubit Turn on -X fields and turn off cluster state coupling Bacon Flammia, arXiv:0912.2098 Adiabatic One-way QC 1 2 3 -X -X -X ... -X -X -X n−1 H = −Zn−1 Xn − Sj n-1 n -X -X Sj = Zj−1 Xj Zj+1 j=2 Suppose we prepare |+ on the first physical qubit Turn on -X fields and turn off cluster state coupling Bacon Flammia, arXiv:0912.2098 Adiabatic One-way QC 1 -X 2 -X 3 -X ... -X -X -X n−1 H = −Zn−1 Xn − Sj n-1 n n -X -X H |+ Sj = Zj−1 Xj Zj+1 j=2 Suppose we prepare |+ on the first physical qubit Turn on -X fields and turn off cluster state coupling Bacon Flammia, arXiv:0912.2098 Adiabatic One-way QC 1 -X 2 -X 3 -X ... -X -X -X n−1 H = −Zn−1 Xn − Sj n-1 n n -X -X H |+ Sj = Zj−1 Xj Zj+1 j=2 Suppose we prepare |+ on the first physical qubit Turn on -X fields and turn off cluster state coupling Rotating the X fields in X-Y plane to make it universal Bacon Flammia, arXiv:0912.2098 Adiabatic One-way QC 1 -X 2 -X 3 -X ... -X -X -X n−1 H = −Zn−1 Xn − Sj n-1 n -X -X U |+ Sj = Zj−1 Xj Zj+1 j=2 Suppose we prepare |+ on the first physical qubit Turn on -X fields and turn off cluster state coupling Rotating the X fields in X-Y plane to make it universal The gap is still constant Bacon Flammia, arXiv:0912.2098 Classical Transistors An “identity gate” Problem: quantum information cannot be cloned Quantum Transistors? 1. Many-body system in its ground state Bacon Crosswhite Flammia, in preparation Quantum Transistors? |ψ 1. Many-body system in its ground state 2. Qubits localized on one side of the device Bacon Crosswhite Flammia, in preparation Quantum Transistors? |ψ 1. Many-body system in its ground state 2. Qubits localized on one side of the device 3. Apply a strong 1-qubit external field to device Bacon Crosswhite Flammia, in preparation Quantum Transistors? |ψ U |ψ 1. Many-body system in its ground state 2. Qubits localized on one side of the device 3. Apply a strong 1-qubit external field to device 4. Qubits now localized on other side of device with a quantum circuit applied to the qubits Bacon Crosswhite Flammia, in preparation Adiabatic Quantum Transistors 1 2 3 ... n-1 What if we turn on the fields all at once? n Adiabatic Quantum Transistors 1 2 3 -X -X -X ... -X -X -X n-1 -X n -X H(t) = (1 − t)HC + tHX This is the transverse-field Ising model (with funny BCs) The gap is = Θ(1/n) In analogy with transistors: An applied field induces a quantum phase transition between an insulating and a “quantum logic” phase. Quantum Transistor Dictionary R(θ) = exp(−iθZ/2) Example How slowly must we turn on the external field in order for the device to successfully quantum compute? Gap Scaling: 1D Numerics Energy gap (!) 1 Mean and standard deviation Fit of 4.97n-1 Smallest and largest gaps Non-twisted 0.1 0.01 10 100 Number of qubits (n) Gap for 1D circuit with random angle rotations... Gap Scaling: 2D Numerics Energy gap (!) 1 Untwisted Random angles Fit of 1.63n-1 0.1 0.01 10 Number of qubits (n) Gap for 2D circuit with random angle rotations Gap Scaling Summary How slowly must we turn on the external field in order for the device to successfully quantum compute? In 1D with no twists, we have rigorously proven the gap is inverse polynomial in the circuit size In 1D with twists, we have extremely strong evidence of polynomial scaling in the circuit size For the 2D case with twists, we have some evidence of polynomial scaling in the size of circuit We have not yet simulated the case with gadgets. Fault-Tolerance Energy 1D untwisted Hamiltonian decouples into two 1D Ising chains with a transverse field. 2.0 1.5 1.0 0.5 0.2 0.4 0.6 0.8 1.0 s Two types of errors: • Those that change energy • Those within the degeneracy • Includes system-bath couplings and Hamiltonian perturbations Assume the excitations obey detailed balance and are suppressed by a Boltzmann factor. Adiabatic evolution preserves eigenstates, so excitations can be (mathematically) dragged back to the beginning. Straightforward stabilizer arguments shows that these are correctible local and independent Pauli errors Fault-Tolerance Energy 1D untwisted Hamiltonian decouples into two 1D Ising chains with a transverse field. 2.0 1.5 1.0 0.5 0.2 0.4 0.6 0.8 1.0 s Two types of errors: • Those that change energy • Those within the degeneracy • Includes system-bath couplings and Hamiltonian perturbations Quantum info is susceptible to decoherence near the beginning and end, but in the middle string-like stabilizer operators give us topological protection from local errors. We can reschedule the adiabatic evolution so that we only spend a constant amount of time in the bad regime, and these errors are local and independent there. Conclusion Adiabatic Gate Teleportation A B |Φ |ψ can be combined with cluster states to build robust adiabatic quantum logic elements |ψ U |ψ Energy 2.0 1.5 1.0 0.5 0.2 0.4 C 0.6 0.8 1.0 s
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Canadian Summer School on Quantum Information (CSSQI) (10th : 2010)
Adiabatic Quantum Transistors Flammia, Steven Jul 23, 2010
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Title | Adiabatic Quantum Transistors |
Creator |
Flammia, Steven |
Contributor | University of British Columbia. Department of Physics and Astronomy Workshop on Quantum Algorithms, Computational Models and Foundations of Quantum Mechanics Pacific Institute for the Mathematical Sciences |
Date Issued | 2010-07-23 |
Description | The invention of the transistor was a watershed moment in the history of computing: it provided a logic element that was naturally robust to noise and error. Quantum computers offer the potential to exponentially speed up some computational problems, but have not been built in large part because quantum information is notoriously fragile and quickly becomes classical information in the presence of noise. In theory, the quantum threshold theorem asserts that these difficulties can be circumvented, but in practice the requirements of this theorem are daunting. Here we propose a novel method for building a fault-tolerant quantum computer which much more closely mimics the classical transistor. We show how a suitably engineered material can be made to quantum compute by the simple application of an external field to the sample. This construction opens a new path toward the engineering of a large-scale quantum computer with design and control advantages o! ver prior state of the art. Just as a transistor works by causing a phase transition between an insulating and conducting phase conditional on an external electric field, the applied field here causes a phase transition at the end of which a quantum gate has been enacted and quantum information propagated across the device. |
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Language | eng |
Date Available | 2016-11-22 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0103162 |
URI | http://hdl.handle.net/2429/30069 |
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Non UBC |
Peer Review Status | Unreviewed |
Scholarly Level | Postdoctoral |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
Aggregated Source Repository | DSpace |
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