├── README.md ├── spinDown.idraw ├── spinUp.idraw ├── confusedCat.jpg ├── animated-qubits.png ├── interferometer.jpg ├── doubleHadamardOf0.png ├── classicalRandomNot.png ├── interferencePattern.jpg ├── .gitignore ├── jshintConfig.json ├── main.js ├── package.json ├── browserSupportDetection.js ├── gulpfile.js ├── src ├── manageHelpSections.js └── manageAnimatedQubitExamples.js ├── LICENSE ├── spinDown.svg ├── spinUp.svg ├── main.css ├── index.html ├── part2.html └── jquery.min.js /README.md: -------------------------------------------------------------------------------- 1 | An introductory article on Quantum Computing -------------------------------------------------------------------------------- /spinDown.idraw: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/davidbkemp/QuantumComputingArticle/HEAD/spinDown.idraw -------------------------------------------------------------------------------- /spinUp.idraw: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/davidbkemp/QuantumComputingArticle/HEAD/spinUp.idraw -------------------------------------------------------------------------------- /confusedCat.jpg: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/davidbkemp/QuantumComputingArticle/HEAD/confusedCat.jpg -------------------------------------------------------------------------------- /animated-qubits.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/davidbkemp/QuantumComputingArticle/HEAD/animated-qubits.png -------------------------------------------------------------------------------- /interferometer.jpg: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/davidbkemp/QuantumComputingArticle/HEAD/interferometer.jpg -------------------------------------------------------------------------------- /doubleHadamardOf0.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/davidbkemp/QuantumComputingArticle/HEAD/doubleHadamardOf0.png -------------------------------------------------------------------------------- /classicalRandomNot.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/davidbkemp/QuantumComputingArticle/HEAD/classicalRandomNot.png -------------------------------------------------------------------------------- /interferencePattern.jpg: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/davidbkemp/QuantumComputingArticle/HEAD/interferencePattern.jpg -------------------------------------------------------------------------------- /.gitignore: -------------------------------------------------------------------------------- 1 | lib-cov 2 | *.deps.js 3 | *.seed 4 | *.log 5 | *.csv 6 | *.dat 7 | *.out 8 | *.pid 9 | *.gz 10 | *.iml 11 | .idea 12 | 13 | pids 14 | logs 15 | results 16 | 17 | node_modules 18 | bower_components 19 | -------------------------------------------------------------------------------- /jshintConfig.json: -------------------------------------------------------------------------------- 1 | { 2 | "eqnull":true, 3 | "eqeqeq":true, 4 | "forin":true, 5 | "immed":true, 6 | "latedef":true, 7 | "newcap":true, 8 | "noarg":true, 9 | "nonew":true, 10 | "regexp":true, 11 | "undef":true, 12 | "unused":true, 13 | "strict":true, 14 | "trailing":true 15 | } -------------------------------------------------------------------------------- /main.js: -------------------------------------------------------------------------------- 1 | /*global jQuery, require */ 2 | 3 | (function () { 4 | "use strict"; 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4 | 5 | function supportsSvg() { 6 | try { 7 | return document.implementation.hasFeature("http://www.w3.org/TR/SVG11/feature#BasicStructure", "1.1"); 8 | } catch (error) { 9 | return false; 10 | } 11 | } 12 | 13 | function supportsES5() { 14 | return [].forEach != null && Object.keys != null; 15 | } 16 | 17 | if (!supportsSvg() || !supportsES5()) { 18 | alert("It seems that your browser does not support some features required by this site. 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Google Chrome)"); 19 | } 20 | 21 | })(); 22 | -------------------------------------------------------------------------------- /gulpfile.js: -------------------------------------------------------------------------------- 1 | var gulp = require('gulp'); 2 | var gutil = require('gulp-util'); 3 | var browserify = require('gulp-browserify'); 4 | var uglify = require('gulp-uglify'); 5 | var jshint = require('gulp-jshint'); 6 | 7 | var scripts = ["main.js", "browserSupportDetection.js", "src/**/*.js"]; 8 | 9 | gulp.task('browserify', function() { 10 | gulp.src('main.js') 11 | .pipe(browserify()) 12 | .pipe(uglify()) 13 | .pipe(gulp.dest("dist")); 14 | }); 15 | 16 | gulp.task('hint', function() { 17 | gulp.src(scripts) 18 | .pipe(jshint('jshintConfig.json')) 19 | .pipe(jshint.reporter('default')); 20 | }); 21 | 22 | gulp.task('watch', ['default'], function () { 23 | gulp.watch(scripts, ['default']); 24 | }); 25 | 26 | gulp.task('default', ['browserify', 'hint']); -------------------------------------------------------------------------------- /src/manageHelpSections.js: -------------------------------------------------------------------------------- 1 | /*global jQuery, module */ 2 | 3 | (function () { 4 | "use strict"; 5 | 6 | function manageHelpSection() { 7 | /*jshint validthis:true */ 8 | var helpElement, helpContent, helpButton; 9 | 10 | function showHelp() { 11 | helpElement.html(helpContent); 12 | } 13 | 14 | helpElement = jQuery(this); 15 | helpContent = helpElement.html(); 16 | helpButton = jQuery(" 120 | 123 | 124 | 125 |
State: 126 | 127 |
128 | 129 | 130 | 131 | 132 | 133 | 134 |

Random NOT

135 |

Random NOT: A NOT operator that has a specified chance of flipping a bit.

136 |

137 | Although not very common, 138 | the “Random NOT” is still just a classical (non-quantum) operator, 139 | but it will help me explain the workings of some quantum operators. 140 |

141 |

142 | Consider applying a Random NOT twice to a bit whose initial value is 0, 143 | where the operator has, for instance, a 30% chance of flipping the bit. 144 | What is the probability of the final state being 0? 145 |

146 |

147 | There are a couple of possible scenarios. 148 | For instance, the first Random NOT might flip the bit from 0 to 1, 149 | and the second Random NOT might flip the bit back to 0. 150 | We represent this as: 151 |

152 |

153 | 0 → 1 → 0 154 |

155 |

There are two paths leading to a final state of 0:

156 | 164 |

And so the final state will be 0 with a probability of 49% + 9% = 58%

165 | Probability tree for random NOT 166 | 167 | 168 |

Random NOT (your turn)

169 |

Next we provide an interactive animation of the Random NOT operator.

170 |

The blue disk now splits in two so that we can track the different possible outcomes.

171 |

The probability of being in a state is represented by the radius of the disk.

172 |

173 | Press the “Random NOT” button multiple times and 174 | note how the arrows add head to tail. 175 |

176 |

177 | Still nothing quantum mechanical about any of this. 178 | We are still just warming up. 179 |

180 | 181 |
182 |
183 |
184 | 187 |
188 |
189 |
State: 190 | 191 |
192 | 193 |
194 | 195 | 196 | 197 | 198 |

Measurement

199 |

200 | We have seen how a random NOT operator can 201 | cause a conventional computer to have various probabilities 202 | of being in different states. 203 | Of course 204 | in reality it is in only one of those states. 205 | We just don't know which one. 206 | Strangely, this is an assumption about reality that we will need to reconsider 207 | when we look at qubits. 208 |

209 |

210 | If you peek at the system to determine its actual state, 211 | then the probabilities all collapse 212 | so that one state (the observed state) 213 | is deemed to now have a probability of 1, 214 | and all the others are deemed to have a probability of 0. 215 |

216 |

217 | Remember, the larger the blue disk, 218 | the more likely the system will turn out to be in that state. 219 |

220 |

221 | In quantum computing, 222 | the word measurement refers to this act of peeking. 223 |

224 |

225 | Press the “Random NOT” button multiple times 226 | and then press the “measurement” button. 227 |

228 |

229 | Note that there is still nothing quantum mechanical about this yet. 230 | That comes next! 231 |

232 | 233 |
234 |
235 |
236 | 239 |
240 |
241 | 242 |
243 |
244 |
State: 245 | 246 |
247 | 248 |
249 | 250 | 251 | 252 | 253 |

Hadamard of 0

254 |

255 | The “Hadamard operator” is 256 | a special quantum operator that can be applied to qubits. 257 |

258 |

259 | Warning: this first look at quantum operators will be pretty boring. 260 | I promise it will get interesting soon! 261 |

262 |

As you will see below, the Hadamard initially acts like a Random NOT with 50% chance of success.

263 |

264 | In this interactive example, 265 | I purposely disable the Hadamard button after you press it. 266 | Later in this article we will see what happens when you apply the Hadamard twice in a row. 267 |

268 | 269 | 270 |
271 |
272 |
273 | 277 |
278 |
279 | 280 |
281 |
282 |
State:
283 | 284 |
285 | 286 | 287 | 288 |

Nothing unusual about that was there?

289 |

But you will be surprised by what comes next...

290 | 291 | 292 | 293 | 294 |

Hadamard of 1

295 |

Things start to become weird when you look at the Hadamard of 1.

296 |

Look carefully at the arrow directions.

297 |
298 |
299 |
300 | 304 |
305 |
306 | 307 |
308 |
309 |
State:
310 | 311 |
312 | 313 | 314 | 315 |

Huh?

316 | Photograph of confused cat 317 | 318 | 319 | 320 |

Phase

321 |

322 | Puzzled? 323 | You should be if this is all new to you. 324 | Please hang in there for a while longer. 325 |

326 | 327 |

328 | The arrow directions represent what physicists call phase: 329 |

330 | 331 | 336 | 337 |

338 | In the case of nuclear spin, 339 | phases can be manipulated by applying electric and/or 340 | magnetic fields. 341 |

342 |

343 | We will see the importance of phase in a moment, 344 | but first let's look at another interesting quantum computing operator... 345 |

346 | 347 | 348 | 349 |

T Operator

350 |

The T operator rotates the phase of 1, but leaves 0 untouched.

351 |

Note how it does not affect the probabilities at all.

352 |
353 |
354 |
355 | 359 |
360 |
361 |
State:
362 | 363 |
364 | 365 | 366 | 367 | 368 |

Measurement Revisited

369 |

Recall:

370 | 378 |

379 | Important: 380 | The likelihood of a state being observed 381 | is entirely determined by the size of the blue disk, 382 | and is completely unaffected 383 | by the direction of the arrow. 384 |

385 |
386 |
387 |
388 | 389 |
390 |
391 | 392 |
393 |
394 |
State:
395 | 396 |
397 | 398 | 399 | 400 |

Quantum Interference

401 | 402 |

Consider what happens when we apply a Hadamard operation twice in a row.

403 | 404 |

405 | Let's assume that a qubit is initially known to definitely have the value 0. 406 | If you were to apply the Hadamard to it twice in a row, 407 | then there are four equally likely scenarios 408 | (Recall that “x → y → z” 409 | means “the qubit starts with a value x, 410 | the first Hadamard results in the qubit having the value y, 411 | and the second Hadamard results in the qubit having the value z”): 412 |

413 | 419 | 420 |

421 | So the final value should be equally likely to be 0 or 1 but, 422 | in reality, 423 | applying the Hadamard operator twice in a row 424 | always returns the qubit to its original value. 425 | In our case, where the qubit is initially 0, 426 | two applications of the Hadamard will result in it being 0 again. 427 |

428 | 429 |

430 | Try it out. 431 | Press the “Apply Hadamard” button twice 432 | and watch it return to having a 100% likelihood of having the value 0. 433 |

434 | 435 |
436 |
437 |
438 | 439 |
440 |
441 |
State:
442 | 443 |
444 | 445 | 446 | 447 |

Totally confused?

448 |
449 | If quantum mechanics hasn't profoundly shocked you, you haven't understood it yet. 450 |
451 | Niels Bohr 452 |
453 | 454 | 455 | 456 |

What is going on here?

457 | 458 |

459 | The state of the qubit after the first Hadamard 460 | seems to have a 50% chance of being 0 461 | and a 50% chance of being 1. 462 |

463 |

464 | The second Hadamard is applied to both the 0 and 1 states 465 | and the results are combined. 466 |

467 | 468 |

469 | The arrows still add head to tail. 470 |

471 | 472 |

473 | 474 | The two different scenarios ending in a 1 state have opposite phases 475 | and so they cancel each other out. 476 | 477 |

478 |

479 | This process of phases causing possible outcomes to cancel or re-enforce 480 | is what physicists call interference. 481 |

482 | 483 |

This is what philosophers of physics loose sleep over.

484 | 485 | Paths for applying Hadamard twice to 0. 486 | 487 |

488 | By the way, 489 | the mathematically inclined may be worried about 490 | all the probabilities not adding up to 1 any more. 491 | The trick is that 492 | the arrow lengths now have to represent 493 | the square roots of the probabilities. 494 | We will briefly cover this in more detail in the section entitled 495 | Some mathematics 496 | in Part 2. 497 |

498 | 499 | 500 | 501 | 502 |

Hadamard of 1 (revisited)

503 |

504 | It is instructive to observe the effects of applying 505 | a Hadamard twice in a row when the initial value is 1. 506 | This time, the qubit returns to 1: 507 |

508 |
509 |
510 |
511 | 512 |
513 |
514 |
State:
515 | 516 |
517 | 518 | 519 | 520 |

Different kinds of uncertainty

521 | 522 |

523 | We are actually dealing with two different kinds of uncertainty: 524 |

525 | 526 | 540 | 541 | 542 | 543 |

Small Diversion: Superposition of Locations

544 | 545 |

546 | So far, the rather abstract phenomenon of nuclear spin is 547 | the only approach that I have mentioned for creating qubits. 548 |

549 |

550 | Quantum physics seems even more bizarre 551 | when you discover that physical objects can be 552 | in superpositions of different locations. 553 |

554 |

555 | The photons travelling through an “interferometer” 556 | are in superpositions of locations that can be kilometres apart 557 | (as they are in the 558 | LIGO 559 | interferometer). 560 |

561 |

562 | A simple interferometer is shown below. 563 | Photons are emitted by a light source 564 | (e.g. a laser) 565 | that is pointing at a 566 | “half silvered mirror”, 567 | which reflects some of the light and lets some of the light through. 568 |

569 | Interferometer 570 |

571 | Individual photons end up in a superposition of 572 | having been reflected 573 | and having been let through. 574 | A couple more mirrors are used to bring the split light beam back together 575 | at a detector. 576 | The positions of the mirrors and the detector all effect the lengths of 577 | the two different paths, 578 | so that one path can be longer than the other. 579 | Like the T operator described earlier, 580 | a change in the relative path lengths will alter 581 | the relative phases of the two photon states. 582 | A difference equal to the wavelength of light is enough to 583 | change the relative phases by an entire 360 degrees. 584 | If the phases are exactly opposite, 585 | then they will cancel each other out, 586 | and the detector will not detect anything. 587 | The resulting effect will be an alternating series of light 588 | and dark concentric rings like those shown below. 589 |

590 | 591 | 592 |

593 | This interference effect even happens when 594 | the light source is slowly emitting photons 595 | one at a time. 596 |

597 | 598 | 599 | Interference Pattern 600 | 601 |

602 | It is tempting to think that the half silvered mirror 603 | is splitting each photon in two 604 | and that the interference effects are caused by the two photons 605 | interacting with each other. 606 | But this is not what happens. 607 |

608 |

609 | If detectors are placed on the two paths, 610 | and the light source is slowly emitting photons 611 | one at a time, 612 | then the detectors only ever detect a photon 613 | on one path or the other. 614 | They never detect two photons at once! 615 | (Well, they very occasionally do due to the light source 616 | very occasionally emitting two at once, 617 | but the frequency that this should happen is 618 | easily predicted and verified.) 619 |

620 | 621 |

622 | If detectors are placed on either or both of the two paths, 623 | then the act of detecting the presence (or absence) of the photon 624 | causes the superposition to collapse to one or the other, 625 | and the interference effects disappear, 626 | even if the detector lets the photon continue on. 627 |

628 | 629 | 630 | 631 |

632 | It gets even more interesting when you have more than one qubit 633 |

634 | 635 |

636 | The quantum weirdness rises to a whole new level when there are two or more qubits interacting. 637 | This is explored in Part 2. 638 |

639 |

640 | If you want to experiment with various single qubit quantum operations first, 641 | then have a play with the 642 | 643 | Quantum Computer Gate Playground 644 | 645 |

646 | 647 | 648 |
649 |

Attributions

650 |

Michelson Interferometer: http://commons.wikimedia.org/wiki/File:Michaelson_with_letters.jpg

651 |

Interference Pattern: http://commons.wikimedia.org/wiki/File:Zonenplatte_Cosinus.png

652 |
653 | 654 | 657 | 658 | 659 | 660 | 661 | 662 | 663 | 673 | 674 | 675 | -------------------------------------------------------------------------------- /part2.html: -------------------------------------------------------------------------------- 1 | 2 | 3 | 4 | 5 | 6 | An Interactive Introduction To Quantum Computing Part 2 7 | 8 | 9 | 10 | 11 |
12 | 13 |

An Interactive Introduction To Quantum Computing Part 2

14 |
15 | Quantum Search 16 |
17 | 18 |

19 | This article was originally written in 2014, 20 | but has had some minor improvements in December 2017 and January 2018 21 | when I received some useful feedback after the article 22 | had unexpectedly been posted on 23 | Hacker News 24 | in December 2017. 25 |

26 | 27 |

28 | This is Part 2 of a two part series. 29 | In Part 1 30 | I explained some of the basics of qubits, 31 | including the mysterious notions of phase and quantum interference. 32 | Here in Part 2, you will learn 33 | how quantum computers may one day be able to solve search problems 34 | faster than conventional computers. 35 |

36 | 37 |

Multiple qubits

38 |

39 | Important: There is something really important I feel the need to emphasize 40 | as we start dealing with multiple qubits. 41 |

42 |

The blue disks introduced in Part 1 do not correspond to single qubits.

43 |

44 | Each blue disk corresponds to a distinct state of the entire quantum system. 45 | If you are not sure what I mean by that, 46 | then please hang in there, 47 | it will hopefully become aparent very soon. 48 |

49 | 50 | 51 | 52 |

Controlled NOT

53 |

54 | The Controlled NOT operator is a very simple but powerful 55 | two-qubit operator. 56 |

57 |

58 | The Controlled NOT operator modifies the value of one qubit (the target qubit) 59 | based on the value of another qubit (the control qubit). 60 |

61 |

62 | Specifically, the Controlled NOT operator flips the value of the target qubit 63 | if the control qubit has a value of 1. 64 |

65 |

66 | For example: 67 | Suppose we have two qubits with 68 | an initial value of “11”. 69 | Let's use the left qubit as the control qubit 70 | and the right qubit as the target. 71 | Then the result of applying a Controlled NOT 72 | will be “10”. 73 | As usual, we represent this as 74 | “11 → 10” 75 |

76 |

77 | Continuing with our choice of left qubit as control qubit 78 | and right qubit as target, 79 | there are four scenarios to consider for 80 | the Controlled NOT operator. 81 |

82 | 88 | 89 | 90 | 91 |

Quantum Entanglement

92 |

93 | A combination of the Hadamard followed by a Controlled NOT 94 | can put two qubits into a state of “quantum entanglement”. 95 |

96 |

97 | In what follows, we take a two qubit system in the state “00”, 98 | apply the Hadamard operation to the left qubit, 99 | and then use that left qubit as the control in a Controlled NOT 100 | of the right qubit: 101 |

102 |

Initial State

103 |
104 |
State:
105 | 106 |
107 | 108 |

After applying a Hadamard to the left qubit

109 |

110 | When you apply a Hadamard to the left qubit, 111 | that qubit goes into 112 | a superposition of 0 and 1. 113 | The right qubit is not touched and continues to have the value 0. 114 | So the pair of qubits goes into 115 | a superposition of 00 and 10. 116 |

117 |
118 |
State:
119 | 120 |
121 |

After a Controlled NOT of the right qubit controlled by the left qubit

122 |

123 | The Controlled NOT will leave the 00 state unchanged as the control qubit (left qubit) is 0. 124 |

125 |

126 | When acting on the 10 state, the control qubit is 1, 127 | and so the target bit is flipped resulting in 11. 128 | So the pair of qubits goes into 129 | a superposition of 00 and 11. 130 |

131 |
132 |
State:
133 | 134 |
135 |

136 | Note that, if you were to measure the qubits while they are in this state, 137 | you would find that they are either both 0 or both 1. 138 | This sort of correlation between qubit values 139 | within a superposition is called 140 | “quantum entanglement”. 141 |

142 |

143 | Why quantum entanglement is so special deserves an entire article of its own. 144 | For now, I am simply using entanglement to 145 | emphasise how quantum superposition can apply to 146 | the combined state of a number of qubits. 147 |

148 | 149 | 150 | 151 |

Arbitrary Logic Functions

152 |

153 | In addition to the NOT operator that we have already seen, 154 | a quantum computer can implement the standard 155 | logic functions of AND and OR. 156 | From these, one can compute any function over a fixed number of bits. 157 |

158 |

159 | The details are surprisingly fascinating 160 | and not as straight forward as you might expect. 161 | However, 162 | for the purposes of this article, 163 | I want you to trust me when I say it can be done. 164 | This will allow us to move more quickly to 165 | the even more fascinating topic of quantum search. 166 |

167 |

168 | Before we can cover quantum search, 169 | there is one more quantum operator I need to introduce. 170 |

171 | 172 | 173 | 174 |

Controlled Phase Flip

175 |

176 | Recall that the term phase, 177 | introduced in Part 1, 178 | refers to the arrow direction of a state. 179 |

180 |

181 | The controlled phase flip is a multi qubit operator. 182 | It flips the phase of the states where 183 | all of the specified qubits have the value of 1. 184 |

185 | 186 |

187 | More useful to us, however, 188 | is a generalised form of the controlled phase flip. 189 | It will flip the phase of the states where the target qubits 190 | have any specified combination of 1's and 0's. 191 |

192 | 193 |

Try it yourself:

194 | 195 |
196 |
197 |
198 | 201 | 204 | 207 | 210 |
211 |
212 | 213 |
214 | 215 | 216 | 217 | 218 | 219 |

Quantum Search

220 |

221 | Imagine you are trying to crack an n-bit encryption key by brute force: 222 | i.e. by trying every one of the 2n possible values. 223 |

224 |

225 | This is a classic needle in a haystack search problem. 226 |

227 |

228 | In other words, 229 | it belongs to a class of problems for which 230 | there exists an efficient function, 231 | called an oracle (or verification) function, 232 | that takes a candidate value as input 233 | and returns true if the candidate is the correct value. 234 | However, 235 | the only way to find the correct value 236 | is to test the oracle function on practically every possible value. 237 |

238 |

239 | A conventional computer will need to evaluate the oracle function, 240 | on average, 241 | 2n-1 times. 242 |

243 |

244 | By using 245 | Grover's algorithm, 246 | a quantum computer only needs to evaluate the oracle function less than 247 | √(2n) times. 248 |

249 |

250 | Even if a conventional computer can compute the oracle function in a nanosecond, 251 | if the input was 64 bits long, 252 | then it would probably take hundreds of years 253 | for that conventional computer 254 | to find the answer. 255 | A quantum computer might possibly take just a few seconds. 256 | 257 | A conventional computer would have to evaluate the oracle function an average of 258 | 263 times, 259 | compared to just over four billion times for a quantum computer. 260 | It is likely however, 261 | at least in the early days of quantum computing, 262 | that individual operations on a quantum computer may be vastly slower 263 | than on a conventional computer. 264 | It may take a while before quantum computers ever get 265 | to solve a problem any faster than conventional computers. 266 | 267 |

268 |

269 | For the extremely simple case where the input is only 2 bits, 270 | there are four possible answers. 271 | A conventional computer may need to test up to three 272 | of those possible values. 273 | (If it isn't the first three, 274 | then you can infer it must be fourth one without testing it.) 275 | As you will see below, 276 | a quantum computer can find the answer 277 | using a single invocation of the oracle function. 278 |

279 | 280 | 281 | 282 |

Quatum Search over 2 Qubits

283 | 284 |

285 | First recall that 286 | the objective is to find the value of x 287 | for which the oracle function is true. 288 | For Grover's Search Algorithm to work, 289 | it is essential that the oracle function is true 290 | for exactly one value of x. 291 |

292 | 293 |

294 | In the conventional descriptions of Grover's Search Algorithm, 295 | it is assumed that the the oracle function 296 | will flip the value of an “output” qubit 297 | when the input matches the desired result. 298 | It is a lot more convenient if we do away with the output bit, 299 | and instead flip the phase 300 | when the input matches the desired result. 301 |

302 | 303 |

304 | Here are the steps to Grover's Search Algorithm 305 | for the very simple case of two bits. 306 |

307 |

308 | For example, the problem may be to 309 | find factors of 15 that are less than 4. 310 | This is obviously trivial for such small values, 311 | but the problem of finding factors for very large numbers 312 | (e.g. numbers with over 100 digits) 313 | is currently a very computationally expensive problem. 314 |

315 | 316 |
    317 |
  1. 318 | Start with the qubits in a superposition of all possible states: 319 | 00, 01, 10, and 11 (all with the same phase). 320 |
    2. 321 |
  2. 322 | Apply the oracle function once only. 323 | Note that, 324 | because the system starts in a superposition of all possible states, 325 | the oracle function is effectively applied to all possible states, 326 | but it will only flip the phase of the state for which we are searching. 327 | If you were to perform a measurement now, 328 | all the possible outcomes are still equally likely 329 | and so we don't seem to have made any progress. 330 | However, the next three steps use quantum interference to find the correct answer. 331 |
    3. 332 |
  3. Apply a Hadamard operation to both qubits.
    4. 333 |
  4. Flip the phase of the 00 state.
    5. 334 |
  5. Apply a Hadamard operation to both qubits again.
    6. 335 |
  6. 336 | Constructive and destructive quantum interference will ensure that, 337 | of the four possible states, 338 | a measurement is guaranteed to result in only the desired state. 339 |
    7. 340 |
341 | 342 |

343 | You can try it yourself below. 344 | Your task is to find the value for which the oracle function 345 | f 346 | is "true". 347 | i.e. the state for which 348 | f 349 | will flip the phase. 350 | On a classical computer you would typically end up testing 351 | at least half the states before finding the answer. 352 | On a quantum computer you can test all the states simultaneously. 353 |

354 | 355 |
357 |
358 |
359 |
360 | Apply the oracle function (to all states): 361 | 364 |
365 | 366 |
367 | 369 | 371 |
372 |
373 | 375 |
376 |
377 | 379 | 381 |
382 |
383 | Only the searched-for state now has a non-zero probability 384 | and a measurement can be safely applied. 385 |
386 |
387 |
388 | 389 |
390 | 391 |
392 |
393 | 394 | 395 |

396 | Don't forget, 397 | unlike what happens in the animation, 398 | each operation is effectively applied to 399 | all the states simultaneously. 400 |

401 | 402 | 403 | 404 | 405 |

Beyond Quantum Search

406 |

407 | The two bit search is a bit of a special case 408 | where the oracle function only needs to be invoked once. 409 | As the number of bits increases, 410 | it becomes necessary to invoke the oracle function multiple times. 411 | In fact it needs to be invoked 412 | ¼π√(2n) 413 | times. 414 |

415 |

416 | While general quantum search techniques may seem quite impressive, 417 | even ¼π√(2n) gets very large very quickly. 418 |

419 |

420 | All you have to do is double the number of bits to wipe out any advantages. 421 | For example, 422 | over a 128 bit search space, 423 | a quantum computer would need to evaluate the oracle function 424 | almost as many times as a conventional computer does 425 | over a 64 bit search space. 426 | Even at one evaluation per nanosecond, 427 | it would take hundreds of years to crack 428 | a 128 bit encryption key. 429 |

430 |

431 | However, 432 | encryption keys 433 | commonly used for securing internet traffic 434 | (RSA keys) 435 | can be broken using a special purpose quantum algorithm. 436 | These encryption schemes rely on the difficulty of 437 | factoring very large numbers, 438 | something that can be easily done on a quantum computer using 439 | Shor's Algorithm. 440 |

441 |

442 | Shor's Algorithm has a “asymptotic” behaviour of 443 | O((log N)3). 444 | I won't try to explain 445 | the exact meaning of this statement, 446 | but it is quite possible that 447 | current encryption keys could be broken in sub-second time frames. 448 | Also, simply doubling the length of the key would have a barely noticeable 449 | effect on the time taken. 450 | The limiting factor is likely to be the number of qubits required 451 | since you need at least as many qubits as the key length. 452 |

453 | 454 | 455 | 456 |

D-Wave

457 |

458 | You may have heard of 459 | D-Wave 460 | quantum computers. 461 |

462 |

463 | These are currently the only commercially available quantum computers. 464 | D-Wave are building quantum computers containing 512 qubits, 465 | which in theory could be in a super position of 466 | 2512 states. 467 | 468 | Update Jan 2017: 469 | 470 | D-Wave have now built a 2000 qubit system 471 | and have plans for a 4000 qubit system. 472 | 473 | 474 |

475 |

476 | Despite having such a large number of qubits, 477 | there is little evidence that they have been solving 478 | problems any faster than conventional computers. 479 |

480 |

481 | D-Wave computers are special purpose 482 | “quantum annealing” computers. 483 | The quantum computers that I have been describing so far are 484 | known as “quantum gate” computers. 485 | Describing the differences are beyond the scope of this 486 | (already long) article. 487 |

488 |

489 | Before they can start out-performing conventional computers, 490 | it is possible that D-Wave simply needs to use more qubits, 491 | or perhaps they may need to figure out how to make 492 | more effective use of their existing qubits. 493 | However, 494 | it quite possibly will not be long before 495 | we start seeing impressive results. 496 |

497 | 498 | 499 | 500 |

Some philosophy

501 |

502 | Describing quantum computing 503 | in terms of superpositions of states 504 | that “collapse” 505 | to the observed state 506 | upon “measurement” 507 | is a very conventional approach 508 | to explaining the physics 509 | behind quantum computing. 510 | However, there are some serious 511 | deficiencies with this approach. 512 |

513 |

514 | Probably the most fundamental problem is: 515 | 516 | What is so special about measurement that it causes 517 | quantum superpositions to collapse? 518 | 519 |

520 |

521 | The measuring equipment itself is made of the same stuff 522 | as the qubits being measured 523 | (protons, neutrons, electrons, and photons). 524 |

525 |

526 | When measuring a qubit that is in a superposition of 527 | two different states, 528 | there is nothing in quantum physics that says why 529 | the measuring equipment 530 | might cause the superposition to collapse to a single state 531 | instead of the measuring equipment 532 | becoming entangled with the qubit. 533 | Instead of the quantum state collapsing, 534 | the result could be a superposition of the following two states: 535 |

536 | 540 |

541 | If you consider your conscious awareness to be 542 | nothing more than patterns of activity in your brain, 543 | and that your brain is governed by the laws of physics, 544 | then there is nothing in quantum physics that says why, 545 | when you look at the measurement outcome, 546 | you yourself do not become entangled with 547 | the quantum qubit and 548 | the measuring equipment. 549 | The result would be a superposition of the following two states: 550 |

551 | 557 |

558 | Think of it as two alternative realities existing. 559 | In one reality you see a 0, 560 | and in the other you see a 1. 561 | Quantum interference is the only ways that these realities interact. 562 |

563 |

564 | Variations on this interpretation of quantum physics 565 | are taken quite seriously 566 | by many physicists and physics philosophers, and is 567 | commonly referred to as the “Many Worlds” 568 | interpretation of quantum physics. 569 | Much has been written on the topic, 570 | and an internet search will point you to 571 | many good (and not so good) articles. 572 |

573 |

574 | Quantum computing 575 | provides a nice way of looking at some of these issues. 576 | It may even provide a way of simulating ideas by, 577 | for example, 578 | representing different states of an observer using qubits. 579 |

580 | 581 |

582 | This article has already covered more topics than I originally planned. 583 | However, I feel the need to include a little bit of 584 | a mathematical explanation to some of the material. 585 | For those that are not interested in the mathematics, 586 | feel free to skip to the 587 | Useful Links Section 588 | at the end. 589 |

590 | 591 | 592 |

Some mathematics

593 |

594 | In this article 595 | I used little blue disks with red arrows 596 | to represent the probabilities 597 | of a set of bits or qubits 598 | being in different states. 599 | It is likely that I caused some confusion 600 | by quietly using the disks in subtly different ways 601 | when describing 602 | two quite different types of uncertainty. 603 |

604 |

605 | When describing ordinary non-quantum bits, 606 | the system is in a definite state, 607 | but which state that is may be unknown. 608 | In this situation, 609 | it is most convenient if 610 | the arrow lengths are proportional to the probabilities 611 | of the respective states 612 | so that they can be added head to tail. 613 | If there were two scenarios, S1 and S2, both ending in the same state, 614 | where S1 has a probability of p1 615 | and S2 has a probability of p2, 616 | then the probability of that final state is 617 | p1 + p2. 618 |

619 |

620 | When describing qubits 621 | in a superposition of different states, 622 | it is no longer simply a case of not knowing which state they are in, 623 | the system actually seems to be in multiple states at once. 624 | In this situation, 625 | it is most convenient if 626 | the arrow lengths are proportional to the square roots of the probabilities 627 | of the respective states 628 | (the disk areas now effectively represent probabilities 629 | except for a factor of π). 630 | The arrow directions represent their phases. 631 | When two scenarios end in the same state, 632 | you need to use vector addition to add the arrows. 633 |

634 |

635 | The arrows represent what physicists call “amplitudes”. 636 |

637 |

638 | Conventionally, an amplitude is represented using a complex number 639 | whose value on the complex number plane coincides with the end of our arrow. 640 | If our arrow has a length r and a phase θ, 641 | then it is represented as the complex number: 642 |

643 |

644 | r (cos(θ) + i sin(θ)) 645 |

646 |

647 | If a state had this amplitude, 648 | then the probability of that state being the outcome 649 | of a measurement is r2. 650 |

651 |

652 | When we talk of “adding arrows head to tail”, 653 | we can simply add the corresponding complex numbers using 654 | ordinary complex number arithmetic. 655 |

656 |

657 | Given that the probabilities of all possible outcomes have to add up to 1, 658 | it follows that the sum of the squares of the amplitude magnitudes 659 | also have to add up to 1. 660 | All quantum operators preserve this property. 661 |

662 |

663 | If the amplitudes for the 0 and 1 states of a qubit are 664 | the (possibly complex) numbers 665 | a0 666 | and 667 | a1 668 | respectively, 669 | then applying a Hadamard results in 670 | the amplitudes of 0 and 1 becoming 671 | (1/√2)(a0 + a1) 672 | and 673 | (1/√2)(a0 - a1) 674 | respectively. 675 |

676 |

677 | For the simple case of a qubit being 100% in 678 | the 1 state with a phase of 0, 679 | applying a Hadamard results in 680 | the amplitudes of 0 and 1 being 681 | 1/√2 and -1/√2 respectively. 682 | Notice that if you square and add these together that you get 1. 683 |

684 |

685 | That is already more mathematics as I wanted to cover in this article. 686 | For more, see some of the useful links listed in the next section. 687 |

688 | 689 | 690 | 691 | 692 | 693 | 723 | 724 | 725 | 726 | 728 | 729 | 730 | 731 | 732 |
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