├── .gitignore ├── LICENSE ├── README.md ├── animated-qubits.png ├── browserSupportDetection.js ├── classicalRandomNot.png ├── confusedCat.jpg ├── dist └── main.js ├── doubleHadamardOf0.png ├── gulpfile.js ├── index.html ├── interferencePattern.jpg ├── interferometer.jpg ├── jquery.min.js ├── jshintConfig.json ├── main.css ├── main.js ├── package.json ├── part2.html ├── spinDown.idraw ├── spinDown.svg ├── spinUp.idraw ├── spinUp.svg └── src ├── manageAnimatedQubitExamples.js └── manageHelpSections.js /.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 | -------------------------------------------------------------------------------- /LICENSE: -------------------------------------------------------------------------------- 1 | The MIT License (MIT) 2 | 3 | Copyright (c) 2014 David Kemp 4 | 5 | Permission is hereby granted, free of charge, to any person obtaining a copy 6 | of this software and associated documentation files (the "Software"), to deal 7 | in the Software without restriction, including without limitation the rights 8 | to use, copy, modify, merge, publish, distribute, sublicense, and/or sell 9 | copies of the Software, and to permit persons to whom the Software is 10 | furnished to do so, subject to the following conditions: 11 | 12 | The above copyright notice and this permission notice shall be included in all 13 | copies or substantial portions of the Software. 14 | 15 | THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR 16 | IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 17 | FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE 18 | AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER 19 | LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, 20 | OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE 21 | SOFTWARE. 22 | 23 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | An introductory article on Quantum Computing -------------------------------------------------------------------------------- /animated-qubits.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/davidbkemp/QuantumComputingArticle/b24cccdd70f923b8c16fba293da2d68ee065ad3c/animated-qubits.png -------------------------------------------------------------------------------- /browserSupportDetection.js: -------------------------------------------------------------------------------- 1 | /*global document, alert */ 2 | (function () { 3 | "use strict"; 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. Please try a more modern browser (eg. Google Chrome)"); 19 | } 20 | 21 | })(); 22 | -------------------------------------------------------------------------------- /classicalRandomNot.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/davidbkemp/QuantumComputingArticle/b24cccdd70f923b8c16fba293da2d68ee065ad3c/classicalRandomNot.png -------------------------------------------------------------------------------- /confusedCat.jpg: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/davidbkemp/QuantumComputingArticle/b24cccdd70f923b8c16fba293da2d68ee065ad3c/confusedCat.jpg -------------------------------------------------------------------------------- /doubleHadamardOf0.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/davidbkemp/QuantumComputingArticle/b24cccdd70f923b8c16fba293da2d68ee065ad3c/doubleHadamardOf0.png -------------------------------------------------------------------------------- /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']); -------------------------------------------------------------------------------- /index.html: -------------------------------------------------------------------------------- 1 | 2 | 3 | 4 | 5 | 6 | An Interactive Introduction To Quantum Computing 7 | 8 | 9 | 10 | 11 |
12 | 13 |

An Interactive Introduction To Quantum Computing

14 |
15 | Or What Do You Mean They Can Be Both Zero And One At The Same Time! 16 |
17 | qubit animation 18 |

By David Kemp

19 |

@david_b_kemp

20 | 21 |

22 | This article was originally written in 2014, 23 | but has had some minor improvements in December 2017 and January 2018 24 | when I received some useful feedback after the article 25 | had unexpectedly been posted on 26 | Hacker News 27 | in December 2017. 28 |

29 | 30 |

Heard of quantum computers?

31 |

Heard that they are faster than conventional computers?

32 |

Perhaps you have heard of quantum bits (abbreviated to qubits).

33 |

34 | Maybe you have even heard of the puzzling notion that 35 | qubits can have the values 0 and 1 36 | both at the same time. 37 |

38 |

Let me try to explain what this really means.

39 | 40 |

41 | This is part one of a two part series 42 | for those that want to 43 | learn a little about quantum computing, 44 | but lack the mathematics and quantum physics background required 45 | by many of the introductions out there. 46 | It covers some of the basics of quantum computing, 47 | such as qubits, state phases, and quantum interference. 48 | Part 2 49 | goes on to look at quantum search. 50 |

51 | 52 | 53 | 54 |

Qubits

55 |

56 | I assume you know what plain old ordinary binary bits are. 57 | 58 | Sorry, But I cannot assume you know nothing at all! 59 | 60 |

61 |

62 | Conventional bits are implemented using many different approaches: 63 | e.g. voltages on a wire, 64 | pulses of light on a glass fibre, 65 | etc. etc. 66 |

67 |

Just like bits, qubits have a binary state.

68 |

69 | Qubits represent 0 and 1 using quantum phenomenon like 70 | the nuclear spin direction of individual atoms. 71 |

72 |

73 | E.g. use “clockwise” for 0 and “anti clockwise” for 1. 74 |

75 |
76 |
77 | Spin Down 78 |
0
79 |
80 |
81 | Spin Up 82 |
1
83 |
84 |
85 | 86 | 87 | 88 |

The NOT operator

89 |

90 | Consider the conventional NOT (or bit-flip) operator. 91 | 92 | 0 and 1 can represent logical true and false. 93 | NOT true is false, and NOT false is true. 94 | And so, NOT of 1 is 0, and NOT of 0 is 1. 95 | 96 |

97 |

For example, performing a NOT operation on the right most bit of the binary number 111 98 | flips the target bit and results in 110.

99 |

In what follows, it will be convenient to represent the state of a system by listing all possible states 100 | and placing a blue disk next to the current state.

101 |

102 | Click the button labelled “Not bita” 103 | to apply the NOT operation to the left bit, 104 | and click the button labelled “Not bitb” 105 | to apply the NOT operation to the right bit: 106 |

107 |

108 | There is nothing quantum mechanical about these first few interactive examples. 109 | Their main purpose is to familiarise you with 110 | interactive animations I use in this article. 111 |

112 | 113 |
115 |
116 |
117 | 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 |
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