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Hester Knight
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7 REDMOND SYDNEY, DELFT, COPENHAGEN SANTA BARBARA

8 Basic unit: bit = 0 or 1 Computing: logical operation Basic unit: qubit = unit vector α 0 + β 1 Computing: unitary operation α β = β α

9 Basic unit: bit = 0 or 1 Computing: logical operation Description: truth table Basic unit: qubit = unit vector α 0 + β 1 Computing: unitary operation Description: unitary matrix A B Y XOR gate CNOT gate

10 Basic unit: bit = 0 or 1 Computing: logical operation Description: truth table Direction: Most gates only run forward Copying: Independent copies are easy Noise: Manageable with minimal ECC Storage: n bits hold 1 value from 0 to 2 n 1 Input/Output: Linear Computation: An n-bit ALU performs 1 operation Basic unit: qubit = unit vector α 0 + β 1 Computing: unitary operation Description: unitary matrix Direction: Most gates are reversible (matrices) Copying: Independent copies are impossible Noise: Difficult to overcome. Sophisticated QECC Storage: n qubits can hold 2 n values Input/Output: sub-linear Computation: An n-qubit ALU performs 2 n operations

11 Evolution: ψ = U ψ, this may be realized by a Hamiltonian H = ln U Δt Type Basis U Nam e Sym Type Basis U Nam e Sym Pauli 0, 1 0, i i 0 X Y Controlled Not 00, 01, 10, 11} CNOT (CX) Z Rotation 0, 1 e iπ/2 0, i Z S 00, 01, 10, 11} SWAP s 0, e iπ/4 T Measure 0, 1 Qubit to Bit M 0, 1 Identity 0, e iπ/8 R I Binary Control 0, 1 Conditional Application BC Hadamar d 0, H Restore 0, 1 Bit to Qubit Reset

13 From: Nick Bonesteel talk at KITP UCSB

14 Hours (on Cray XE6)

15 Time to Factor N-bit Number RSA-2048 Challenge Problem Number of bits N

16 Nitrogen fixation s-1000s 100s-1000s

17 Ion traps Superconductors Linear optics NV centers Quantum dots Topological

Comparing quantum architectures IBM sc system five transmon qubits: JJ charge states, shunt capacitors, low sensitivity to charge noise Qubits are connected by microwave resonators Automatic

5% Gate times: Single qubit: ~130 ns Two qubit ~250-450 ns Decoherence times: T 1 ~ 60 μs T 2 * about equal to T 1 Native gate set: CNOTs (ECR ZX-90), constrained to star shape Single qubit: Pauli,

18 Comparing quantum architectures IBM sc system five transmon qubits: JJ charge states, shunt capacitors, low sensitivity to charge noise Qubits are connected by microwave resonators Automatic calibration, twice a day Qubits drift between calibrations Addressing: qubit freqs around GHz (all different) Fidelities: Single qubit readout ~96% Single qubit gate ~99.7% Two-qubit gate ~96.5% Gate times: Single qubit: ~130 ns Two qubit ~ ns Decoherence times: T 1 ~ 60 μs T 2 * about equal to T 1 Native gate set: CNOTs (ECR ZX-90), constrained to star shape Single qubit: Pauli, H, S, T UMD ion trap system five 171 Yb + ions: hyperfine states, low sensitivity to B field, linear Paul trap, laser cooled to motional ground state Qubits are connected through pairs of Raman beams Manual calibration, 2 post-docs, 3 grad students Almost no drift between calibrations Addressing: qubit freqs = GHz (all the same) Fidelities: Single qubit readout ~99.4% Single qubit gate ~99.1% Two-qubit gate ~97.0% Gate times: Single qubit: ~20 μs Two qubit ~250 μs Decoherence times: T 1 ~ several hours T 2 * ~ 0.5 s Native gate set: XX(χ) gate (Molmer-Sorensen) on any pair Single qubit: any rotation [Monroe et al., 2017 arxiv: ]

19 Realizations Lifetimes Gate Speed ECC cost Topological (Majorana) 1 minute Nanoseconds 10 1 Flux Qubit / same Charge Qubit / same Transmon / 10 7 same Ion Trap / slower ECC is extremely painful (no quantum refresh like DRAM) Many can be fabricated with variations on standard semiconductor techniques

21 Define a function to perform entanglement: let EPR (qs:qubits) = H qs; CNOT qs The rest of the algorithm: let teleport (qs:qubits) = let qs' = qs.tail EPR qs'; CNOT qs; H qs M qs'; BC X qs' M qs ; BC Z!!(qs,0,2)

23 3 qubits go to 27

24 Largest we ve done: 14 bits (factoring 8193) 14 Million Gates 30 days Circuit for Shor s algorithm using 2n+3 qubits Stéphane Beauregard

25 As defined in: Circuit for Shor s algorithm using 2n+3 qubits Stéphane Beauregard let op (qs:qubits) = CCAdd a cbs AddA' N bs QFT' bs CNOT [bmx;anc] QFT bs CAddA N (anc :: bs) CCAdd' a cbs // Add a to Φ b // Sub N from Φ a + b // Inverse QFT of Φ a + b N // Save top bit in Ancilla // QFT of a+b-n // Add back N if negative // Subtract a from Φ a + b mod N QFT' bs // Inverse QFT X [bmx] // Flip top bit CNOT [bmx;anc] // Reset Ancilla to 0 X [bmx] // Flip top bit back QFT bs // QFT back CCAdd a cbs // Finally get Φ a + b mod N

27 Qubits Memory Time for one gate kbyte microseconds on a watch MByte milliseconds on smartphone GByte seconds on laptop TByte seconds on supercomputer PByte seconds on top supercomputer EByte minutes on future supercomputer ZByte hours on potential supercomputer? 250 size of visible universe age of the universe

28 Quantum Chemistry H = h pq a p a q h pqrs a p a q a r a s pqrs pq Can quantum chemistry be performed on a small quantum computer: Dave Improving Wecker, Bela Quantum Bauer, Bryan Algorithms K. Clark, for Matthew Quantum B. Chemistry: M. B. Hastings, Matthias Hastings, Troyer Ferredoxin D. Wecker, B. Bauer, (Fe M. Troyer The Trotter Step Size 2 SRequired 2 ) used in many metabolic reactions for Accurate Quantum Simulation of Quantum Chemistry As quantum computing We present technology several David improvements Poulin, M. B. Hastings, improves and to On quantum the the standard Chemical Dave Wecker, Trotter-Suzuki Basis Nathan of Trotter-Suzuki Wiebe, Andrew Errors C. Doherty, in Quantum Matthias Chemistry Troyer Simulation including energy transport in photosynthesis computers with based a small algorithms but non-trivial used in number the simulation of Ryan N > 100 Babbush, of quantum qubits Jarrod chemistry McClean, on Dave Wecker, Alán Aspuru-Guzik, Nathan Wiebe appear feasible a in quantum the near computer. The simulation future the First, question we of modify molecules of possible how is Jordan-Wigner a widely anticipated application of quantum computers. However, applications of transformations recent small quantum computers are implemented studies \cite{wbch13a,hwbt14a} gains importance. to Although reduce their One simulation cost from have of linear cast quantum a shadow chemistry on this is hope one of by the revealing most anticipated that the frequently mentioned or logarithmic application Intractable complexity in the is number in Feynman's on gate of orbitals original a count applications classical of to such proposal a constant. of simulations quantum computer of Our computing, increases with the the scaling number of known of spin upper orbitals bounds N as N8, on the simulating quantum modification systems, does which and not becomes in particular require additional prohibitive the complexity electronic ancilla even of for structure qubits. these molecules algorithms Then, we of modest is daunting. size N 100. Prior work This has study bounded was partly errors based due on to of molecules and demonstrate materials. In how a scaling this many analysis paper, operations of we analyze Trotterization the can the be parallelized, step required terms leading of for the an to norm ensemble of the error of random operator artificial and analyzed molecules. scaling Here, with computational a requirements further linear Assumed we for decrease revisit this one of in quantum the the analysis standard parallel respect and algorithms depth scaling: find to the of instead the number to circuit, ~24 that of the at spin-orbitals. billion the scaling is closer years However, to N6 (Nin 11 we find worst scaling) that case these for real error model bounds can perform quantum cost chemistry of a small molecules on constant a quantum factor we have computer. increase studied, be loose in We number indicating focus by up on to of sixteen that qubits the orders random of ensemble magnitude fails for to some accurately molecules. capture Furthermore, the quantum resources required. required First Thirdly, statistical we paper: to modify properties find the the ground term ~850 numerical of state order real-world of thousand in results a the molecules. Trotter-Suzuki for small years Actual systems scaling to fail solve to may reveal be (N significantly any 9 clear scaling) correlation better than between this due ground molecule twice decomposition, to as large as what significantly averaging effects. current classical reducing We computers state the then error error present can and at solve given number an alternative Trotter- of spin-orbitals. simulation We scheme instead and argue show that that chemical it can properties, exactly. We find Suzuki that while timestep. sometimes such A a problem final improvement outperform requires such about modifies existing as the schemes, a ten-fold maximum the Hamiltonian but nuclear that to this charge possibility a molecule depends and crucially the filling on fraction the details of orbitals, of Second increase in the reduce number errors of qubits introduced the simulated paper: over current by the molecule. ~30 technology, non-zero can be We years decisive Trotter-Suzuki obtain further to the for determining solve timestep. improvements (N 7 the scaling) cost using of a quantum version of simulation. the coalescing Our analysis scheme of required increase All in of these number techniques \cite{wbch13a}; of gates are that validated this scheme can motivates be using coherently numerical is several based strategies on simulation using different to use classical Trotter processing steps for different to further terms. reduce The the method required executed is many and orders detailed Third of magnitude gate we use counts paper: to bound larger. are given the This ~5 Trotter for complexity suggests days realistic step that size molecules. to of simulating a given molecule is efficient, in contrast to the for and solve to estimate (N 5.5 the scaling) necessary number of steps, without requiring approach of \cite{wbch13a,hwbt14a} which relied on exponentially costly classical exact simulation. quantum computation to become useful for quantum additional chemistry quantum resources. Finally, we demonstrate improved methods for state problems, drastic algorithmic improvements will be preparation needed. techniques which are asymptotically superior to proposals in the Fourth paper: simulation ~1 hour literature. to solve (N 3, Z 2.5 scaling)

29 H 2 HF H 2 O NH 3 CH 4 HCl F 2 H 2 S Geometries and molecular models from

30 H = h pq a p a q h pqrs a p a q a r a s pqrs pq

31 Mott Insulators Transition Metal Compounds Cuprates (e.g., High Tc SC) Lanthanides and Actinides Kondo Physics (Low temperature Resistance) from Magnetic Impurities Quantum Dots

32 H hub = UΣ i n i n i tσ <i,j>,σ c iσ c jσ H imp = Un n Σ k,σ t k c σ a bath k,σ + h. c. + H bath Solids have regular structure that can be modeled as lattices The Hubbard model only implements H pp and H pqqp terms This doesn t cover many of the materials we re interested in One can choose a single site in the lattice to model The effect of the rest of the lattice can be modeled in terms of its effect on this site U t k U t H bath

33 Impurity Bath H = t ij a i a j w ijkl a i a j a k a l + V ip ( a i a p + a p a i ) + ε p a p a q ijkl ij ip pq

34 G solver ω ω G k, ω G n ω Δ n (ω) Feedback Model Classical Quantum G solver (ω)= c i ω c j ω

35 E gs = k H FF + θ ij H j + M(H k ) i j Good Bad Feedback Classical Model Quantum

37 After Hyart et al

39 SoLi let EPR (q1 : Qubit) (q2 : Qubit) = H q1 CNOT q1 q2 let Teleport (msg : Qubit) (here : Qubit) (there : Qubit) = EPR here there CNOT msg here H msg if JM "Z" [here] = -1 then X there if JM "Z" [msg] = -1 then Z there

45 Core Compiler Machines Gates to Measurement Measurement to Layout Circuit Simulation QCoDeS SoLi Runtime

48 DistillT: 1Q=65 2Q=100 LogQ=81 Frames=277 Connect Dim Data Rows Phys Qubits Data Teleport Block Tele Par Tele Depth Rect 10x10 All 42 15/9 20/9 4*(9+9)=72 Rect 5x9 All 39 18/13 25/13 4*(13+13)=104 Rect 3x18 Half 39 40/31 40/31 4*(31+31)=248 Diag 3x9 Half 26 15/9 15/9 4*(9+9)=72 Diag 2x18 Half 34 39/24 36/24 4*(24+24)=192

50 Energy Conduction Band Surface has no gap Large Gap = Insulator 0 Valence Band Angular Momentum After Hasan, Kane

51 Energy H TI = න dxψ x 2 2m μ iħv xσ y ψ 0 Angular Momentum After Jason Alicia, Winter 2010 Q Meeting

52 Energy 0 Angular Momentum H TIB = න dxψ x 2 2m μ iħv xσ y gμ BB 2 σz ψ H p wave = H TIB + Δψ ψ + h. c. After Jason Alicia, Winter 2010 Q Meeting

53 Energy μ G = di dv 0 Angular Momentum After Leo Kouwenhoven, Summer 2012 Q Meeting

55 After Stanescu, Lutchyn, Das Sarma

Altland-Zirnbauer Random Matrix Classes (http://arxiv.

d + 8 Integer Quantum Hall Effect (GaAs) Carbon nanotubes TI (BiSb) Superfluid 3 He B After Hasan,

56 Altland-Zirnbauer Random Matrix Classes ( Majorana chain (InSb) p x + ip y (SrRuO 4 ) TI (HgTe) Bott Periodicity: d d + 8 Integer Quantum Hall Effect (GaAs) Carbon nanotubes TI (BiSb) Superfluid 3 He B After Hasan, Kane and Kitaev and Freedman et. al

Simulating 250 interacting qubits requires ~ classical bits! State of N interacting qubits: ~ 2 N bits of info!

Hours (on Cray XE6) Simulating 250 interacting qubits requires ~ classical bits! State of N interacting qubits: ~ 2 N bits of info! Time to Factor N-bit Number RSA-2048 Challenge Problem Number of bits