Canadian Summer School on Quantum Information (CSSQI) (10th : 2010)

Adiabatic Quantum Transistors Flammia, Steven Jul 23, 2010

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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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