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High Performance Quantum Computing Matthias Troyer Matthias Troyer | | 1 Beyond exascale: quantum devices Quantum random numbers perfect randomness Quantum encryption secure communication xÈÊ - / Ê , Ê "*9,/ÊÓääÈÊ- / Ê, ]Ê ° Q * , ÊÓ ä ä È Quantum computers? solve quantum models (R. Feynman) factor integers (P. Shor) break RSA encryption solve linear equations (A. Harrow et al) Analog quantum simulators solve quantum models ... Quantum annealer solve hard optimization problems? Matthias Troyer | | 2 Quantum random numbers Quantum mechanics can give true and perfect random numbers 0 0 photo detectors 1 1. Photon source emits a photon 2. Photon hits semi-transparent mirror 3. Photon follows both paths semi-transparent mirror 4. The photo detectors see the photon only in one place: random selection photon source 5. Record one random bit Matthias Troyer | | 3 Quantum cryptography Exchange a secret key to encrypt a message using entangled photons No eavesdropper can listen without being detected except for the quantum hacker Vadim Makarov (Waterloo) Quantum hacking lab Matthias Troyer | | 4 D-Wave Two quantum annealer Solves NP-complete spin glass problem with up to 512 variables H = ∑ J ij si s j + ∑ hi si + const. ij i with si = ±1 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 2 6 10 14 18 22 26 30 34 38 42 46 50 54 58 62 3 7 11 15 19 23 27 31 35 39 43 47 51 55 59 63 64 68 72 76 80 84 88 92 96 100 104 108 112 116 120 124 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125 66 70 74 78 82 86 90 94 98 102 106 110 114 118 122 126 67 71 75 79 83 87 91 95 99 103 107 111 115 119 123 127 128 132 136 140 144 148 152 156 160 164 168 172 176 180 184 188 129 133 137 141 145 149 153 157 161 165 169 173 177 181 185 189 130 134 138 142 146 150 154 158 162 166 170 174 178 182 186 190 131 135 139 143 147 151 155 159 163 167 171 175 179 183 187 191 192 196 200 204 208 212 216 220 224 228 232 236 240 244 248 252 193 197 201 205 209 213 217 221 225 229 233 237 241 245 249 253 194 198 202 206 210 214 218 222 226 230 234 238 242 246 250 254 195 199 203 207 211 215 219 223 227 231 235 239 243 247 251 255 256 260 264 268 272 276 280 284 288 292 296 300 304 308 312 316 257 261 265 269 273 277 281 285 289 293 297 301 305 309 313 317 258 262 266 270 274 278 282 286 290 294 298 302 306 310 314 318 259 263 267 271 275 279 283 287 291 295 299 303 307 311 315 319 320 324 328 332 336 340 344 348 352 356 360 364 368 372 376 380 321 325 329 333 337 341 345 349 353 357 361 365 369 373 377 381 322 326 330 334 338 342 346 350 354 358 362 366 370 374 378 382 323 327 331 335 339 343 347 351 355 359 363 367 371 375 379 383 384 388 392 396 400 404 408 412 416 420 424 428 432 436 440 444 385 389 393 397 401 405 409 413 417 421 425 429 433 437 441 445 386 390 394 398 402 406 410 414 418 422 426 430 434 438 442 446 387 391 395 399 403 407 411 415 419 423 427 431 435 439 443 447 448 452 456 460 464 468 472 476 480 484 488 492 496 500 504 508 449 453 457 461 465 469 473 477 481 485 489 493 497 501 505 509 450 454 458 462 466 470 474 478 482 486 490 494 498 502 506 510 451 455 459 463 467 471 475 479 483 487 491 495 499 503 507 511 1 Can be built with imperfect qubits Suffers form calibration problems like other analog devices Unknown if this technology can ever scale better than classical devices Matthias Troyer | | 6 What about quantum computers ? ? xÈÊ - / Ê , Ê "*9,/ÊÓääÈÊ- / Ê, ]Ê ° Q * , ÊÓ ä ä È Matthias Troyer | | 7 Classical versus quantum bits ▪ Classical bits can only be either 0 or 1 ▪ Quantum bits can have both values at once, in arbitrary superpositions ψ =α 0 +β 1 ▪ 2 need two complex or three real numbers to describe the state ⎛ α ⎞ ψ =⎜ ⎟ β ⎝ ⎠ ▪ 0 α + β =1 2 ⎛θ ⎞ α = cos ⎜ ⎟ ⎝ 2⎠ ⎛θ ⎞ β = e sin ⎜ ⎟ ⎝ 2⎠ iφ 1 but when measuring (looking) I only ever get 1 bit 0 with probability α 2 1 with probability β 2 Matthias Troyer | | 8 Information content of a quantum register ▪ The state of an N qubit register ▪ needs 2N complex numbers to be represented classically ▪ but when measured only gives N bit of information ▪ Advantages and limitations of quantum computers ▪ Exponential intrinsic parallelism: operate on 2N inputs at once ▪ But very limited readout of only N bits ▪ No-cloning theorem: a quantum register cannot be copied Try to build a cloner for 0 and 1 C0 → 0 0 C1 →1 1 Linearity of quantum mechanics means it will not clone an arbitrary state C (α 0 + β 1 ) → α 0 0 + β 1 1 ≠ (α 0 + β 1 )(α 0 +β 1 ) Matthias Troyer | | 9 necessary. In the following I will discuss a set!of typically used one an " and will then discuss which ones are strictly necessary. A set of common two-qubit gates are controlled gates, consisting 0 of 1 a control qubit an and will then discuss which ones are strictly necessary. (NOT) version CU X of a single qubit 1gate a target qubit.Pauli-X The controlled 0 "U (any of the li ! Single qubit qubit gatesoperation U on the target0qubit above) performs the single −i only if the contr Single qubit gates Pauli-Y ! " Y qubit is set to 1. i 0 ! 0 1 "" ! Quantum gates Pauli-X The quantum circuit(NOT) for such a gate is:X 01 010 1 Pauli-X Pauli-Z(NOT) ! 1 0 " XZ• 00 −1 −i "" ! ! Pauli-Y 01i −i 1 Y the NAND gate is universal ▪ In classical computing 1 0 √ Pauli-Y Hadamard gate ! i 0 " Y UH 2 11 −1 0 "" ! ! Pauli-Z 10 1−1 00 Z gates we need three ▪ In quantum computing Pauli-Z Phase gate ! is " ZS , |01⟩ , |10⟩ , |11⟩} Its matrix representation in a basis {|00⟩ i "" 01 0−1 1 ! ! 1 Hadamard gate ⎛ ⎞ √ H 10 11 2 111 −1 1 √ 1 0 0 0 T gate or gate π/8 gate Hadamard ! " 1 → 0 −1) iπ/4 ( 0 → 0 + 1 ) T 2 ( 01 e−1 ▪ Hadamard gate (flips x and z) ⎜ 0 1 H 2 ⎟ 2 ! 1 0" ! " 0 0 ⎜ ⎟ Phase gate . e−iθ/2 10 00i (8.5 S ⎝ ⎠ 0 0 Rz(θ)gate gate ! Phase Rz(θ) iθ/2 " S U 0 e 0 1 0 i " iπ /4 0 0 ! T gate or π/8 gate iπ/4 α 0 + β101 e→ α 0 +e β 1 T 0 Ato few remarks be useful. The gate is the " quantum analog ! X−iθ/2 phase state) ▪ T gate (applies Timportant gate or1π/8 gatemay The most two-qubit gate isTthe controlled-NOT-gate iπ/4 (CNOT ), which 0 eidentity eto the 0 " and is essential gate. The Hadamard gate (H) squares ! Rz(θ) gate the same as a controlled-X gate. It is typically −iθ/2 Rz(θ) drawneas iθ/2 0 aligned e0 rotation around the y axis, rotating a state with z to x. T Rz(θ) gate Rz(θ) iθ/2 • 0 e called π/8remarks gate since can – up to anX global – be wo A few mayitbe useful. The gate quantum x irrelevant y →isxthe x⊕ y phaseanalog flips) ▪ CNOT gate (conditionally " is essential !gate gate. The Hadamard (H) squares the identity and A few remarks may gate be useful. The Xto is the quantum analog o −iπ/8 e identity 0 with iπ/8 rotation the gate y axis, a state aligned to x. T gate. The around Hadamard (H)rotating squares to the and. isz essential T =e iπ/8 0aligned e with The matrix representation is y itaxis, called π/8 gate the since can rotating – up to an irrelevant global phase be w rotation around a state z to– x. T This choice is not unique and having more gates can make a ▪ ⎛ ⎞an irrelevant global phase – be w called π/8 gate since it can – up to " spin space. ! −iπ/8 1 0 0 0 The Rz gate performs a rotation around the z axis in device more efficient e 0 iπ/8 ⎜ ⎟ "Ry ! . gates. 0 are 1 performed 0T = 0 e⎟ around the x and y ⎜ axis by the Rx and iπ/8 −iπ/8 . 0 e e 0 ⎝ 0 0 T0 =1e⎠ iπ/8 | 10 iπ/8 | . 0 e 0 1 0 around 78 The Rz gate performs 0a rotation the z axis in spin space. Matthias Troyer sch algorithm and its generalization, the Deutsch-Jozsa algorithm are the uantum algorithms that show an advantage over classical algorithms, even e problem they solve is somehow artificial. You are given a binary function n values are either 0 or 1) and know that either the function is constant, or ced, i.e. it is 0 for exactly half the inputs and 1 for the other half. The nd Deutsch-Jozsa algorithm can decide between the two cases with exactly on call.3 The Deutsch algorithm: simplest quantum speedup ▪ Check whether a binary function is constant or not tsch algorithm sch algorithm asks the question to decide whether a binary function of one f :{0,1} → {0,1} able f : {0, 1} → {0, 1} is balanced or constant. Classically one has to make▪twoClassically function calls and determine f (0) calls and f (1) to decide since we need= f(1)? two function are needed: f(0) whether f (0) = f (1). Equivalently we can calculate f (0) ⊕ f (1), where ⊕ nary addition modulo 2. If this value is zero, then f is constant. unction f is given as a quantum algorithm Uf that takes an input state |x⟩|y⟩ ▪ Quantum mechanically only one function call by applying |x⟩|f (x) ⊕ y⟩ then we can determine whether the function is constant in a to both arguments at once ction callfunction with the following algorithm: |0⟩ H |1⟩ H U f x y → x f (x) ⊕ y H Uf the U f (α 0 + β 1 ) 0 → α 0 f (0) + β 1 f (1) tart in a state |0⟩|1⟩ and apply a Hadamard gate to each qubit, giving the + |1⟩)(|0⟩ − |1⟩). Applying the function f we obtain needed to read out the answer as one bit ▪ Smart manipulation " 1! (−1)f (0) |0⟩ + (−1)1f (0)⊕f (1) |1⟩ (|0⟩ − |1⟩). (8.7) 2 ⎡ 1+ (−1) f (0)⊕ f (1) 0 + 1− (−1) f (0)⊕ f (1) 1 ⎤ ⎦ 2⎣ state of the second qubit is constant and the global phase irrelevant we drop and focus on just the first qubit’s state √1 (|0⟩ + (−1)f (0)⊕f (1) |1⟩). Applying ( ) ( ) Matthias Troyer | | 11 What would we use it for? ! ! Surprisingly, this question has not been seriously asked until 2012 ! It needs to be an important problem (killer-app) that we cannot solve on an exascale machine ! Compare to state-of-the-art on beyond-exascale hardware and not an unoptimized code on a single CPU core Matthias Troyer | | 12 Grover search ▪ Search an unsorted database of N entries in √N time ▪ Rare case of provable quantum speedup given an oracle ▪ However, the oracle needs to be implemented! ▪ N-entry database or arbitrary stored data needs at least O(N) hardware ▪ Can perform the same search classically in log(N) time with special purpose hardware ▪ Grover search is only useful if the database need not be stored but can be calculated on the fly! Matthias Troyer | | 13 Shor’s algorithm for factoring ▪ Factoring is a hard problem classically: O(exp(N1/3)) for N bits 53693968364269119460795054153326005186041818389302311662023173188470613584169777981247775554355964649 04452615804209177029240538156141035272554197625377862483029051809615050127043414927261020411423649694 63096709107717143027979502211512024167962284944780565098736835024782968305430921627667450973510563924 02989775917832050621619158848593319454766098482875128834780988979751083723214381986678381350567167 ! = 43636376259314981677010612529720589301303706515881099466219525234349036065726516132873421237667900245 9135372537443549282380180405548453067960658656053548608342707327969894210413710440109013191728001673! * ! 12304864190643502624350075219901117888161765815866834760391595323095097926967071762530052007668467350 6058795416957989730803763009700969113102979143329462235916722607486848670728527914505738619291595079 ▪ Shor’s algorithm is polynomial time on a quantum computer ▪ O(N3) using minimal number of 2N+3 qubits ▪ O(N2) using O(N) qubits ▪ O(N) using O(N2) qubits ▪ Shor’s algorithm suddenly made quantum computing interesting Matthias Troyer | | 14 Shor’s algorithm and encryption ▪ Shor’s algorithm can be used to crack RSA encryption ▪ ▪ assuming 10 ns gate time and minimal number of 2N+3 qubits much faster (seconds) when using more qubits RSA cracked in CPU years Shor 453 bits 1999 10 1 hour 768 bits 2009 2000 5 hours 1000000 10 hours 1024 bits ▪ But use of quantum computers to crack RSA is limited since we can anytime switch to post-quantum encryption ▪ quantum cryptography ▪ lattice based cryptography Matthias Troyer | | 15 Solving linear systems of equations (Harrow et al) ▪ Solve linear systems in log(N) time if ▪ matrix can be computed efficiently and need not be stored ▪ only log (N) bits of the answer are needed ▪ problem is well conditioned ▪ First application: electromagnetic wave scattering ▪ Clader, Jacobs, Sprouse (2013) ▪ crossover compared to classical supercomputers is beyond 1000 years wallclock time lx ly Scattered Field Incident Field Matthias Troyer | | 16 Applications running at scale on Jaguar @ ORNL Applications running at scale on Jaguar @ ORNL (Spring 2011) Domain area Code name Institution # of cores Performance Notes Materials DCA++ ORNL 213,120 1.9 PF PF 1.9 2008 Gordon Bell Prize Winner Materials WL-LSMS ORNL/ETH 223,232 1.8 PF 2009 Gordon Bell Prize Winner Chemistry NWChem PNNL/ORNL 224,196 1.4 PF 2008 Gordon Bell Prize Finalist Materials DRC ETH/UTK 186,624 1.3 PF 2010 Gordon Bell Prize Hon. Mention Nanoscience OMEN Duke 222,720 > 1 PF 2010 Gordon Bell Prize Finalist Biomedical MoBo GaTech 196,608 780 TF 2010 Gordon Bell Prize Winner Chemistry MADNESS UT/ORNL 140,000 550 TF Materials LS3DF LBL 147,456 442 TF 2008 Gordon Bell Prize Winner 149,784 165 TF 2008 Gordon Bell Prize Finalist 147,456 83 TF Seismology SPECFEM3D USA (multiple) Combustion SNL Source: S3D T. Schulthess Matthias Troyer | | 17 Solving quantum chemistry on a quantum computer ▪ A killer-app for quantum computing is solving quantum problems ▪ Design a room-temperature superconductor ▪ Develop a catalyst for carbon sequestration ▪ Develop better catalysts for nitrogen fixation (fertilizer) ▪ These problem need better accuracy than we get by using approximate classical algorithms ▪ exponentially hard classically ▪ polynomial complexity on quantum hardware Matthias Troyer | | 18 Quantum algorithms for quantum chemistry ▪ Assume the fastest imaginable quantum computer and consider smallest problem we cannot solve classically: N=50 orbitals, O(100) qubits Scaling Run time for N=50 State of 12/2013 N10 100 000 years 1/2014 pipelining and optimisations 6/2014 faster convergence parallel with with N9 3 years N5.5 1 hour N1.6 seconds? ▪ Quantum information theorists declare victory proving the existence of polynomial time algorithms ▪ We need HPQC specialists to develop better algorithms! Matthias Troyer | | 19 State of the art of hardware ! ! ! many proposals but few are scalable … Matthias Troyer | | 20 Physical realizations and decoherence ▪ Coupling to the environment easily destroys the quantum superposition ▪ Qubits need to be designed to be well isolated from the environment f ψ =α 0 +β 1 i Parallel evolution providing the quantum speedup. Perturbations from the environment destroy the parallel evolution of the computation 21 Matthias Troyer | | The best qubits: ion traps (universities) ▪ Use the motional states of well isolated ions to encode a qubit ▪ ▪ ▪ ▪ up to 14 qubits coherent for several seconds about 100 gate operations 10 µs gate times Image: group of R. Blatt, Innsbruck ! ▪ Advantages and disadvantages ▪ Well isolated from environment ▪ Relatively slow ▪ Hard to scale beyond O(20) qubits Matthias Troyer | | 22 Superconducting qubits (IBM and universities) ▪ Use superconducting current loops to encode a qubit ▪ State of the art ▪ ▪ ▪ ▪ 5 qubits 100 µs coherence times 10 ns gate times about 100 gate operations ▪ Advantages and disadvantages ▪ ▪ ▪ ▪ scalable fast can be built in semiconductor foundries lots of decoherence in a solid state device Matthias Troyer | | 23 Quantum error correction ▪ Example: Shor’s code protects a against sign or bit flip errors ▪ Needs nine qubits and substantially increases number of gates ▪ Codes need to be recursively iterated to protect against multiple errors ▪ Quantum error correction overhead is at least 1000x more qubits and gates for any reasonable computation Matthias Troyer | | 24 Topological qubits (Microsoft and universities) ▪ Encode the qubit in a topological property of a quantum state ▪ ▪ ▪ ▪ Intrinsic protection due to topology No local noise can decohere the qubit No expensive error correction needed Operations done by “braiding” isolated particles ▪ Most promising: Majorana particles ▪ may exist at ends of superconducting nanowires ▪ hope to confirm their existence within 5 years Ettore Majorana Matthias Troyer | | 25 (Optimistic) road map for quantum computers § First quantum devices exist, but computational power is limited § quantum random numbers § quantum encryption § quantum simulators § quantum optimizers § General-purpose quantum computers will always remain special purpose accelerators for selected tasks § My optimistic road map § Today: 14 qubits allowing up to 100 operations § 2020: a long-time stable quantum bit with quantum error correction § 2025: matching classical CPUs on selected applications § 2030: outperforming any hypothetical classical computer on selected applications § Huge potential for classical spin-off technology (e.g. superconducting) Matthias Troyer | | 26 26 Quantum computing applications ▪ Identifying killer-apps for quantum computing is challenging ▪ ▪ ▪ ▪ ▪ the problem has to be hard enough that it cannot be solved on a PC the problem has to be amenable to quantum acceleration the crossover scale has to be short enough to make it useful CMOS technology is a tough competitor we need to consider special purpose classical devices as competitors ▪ Potential applications ▪ ▪ ▪ ▪ factoring and code breaking (limited use) quantum chemistry and material simulations (challenging but enormous potential) solving linear systems (can’t we solve them well enough already?) machine learning??? ▪ It is time to move quantum computing research from theoretical computer science to high performance computing Matthias Troyer | | 27