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Q U A N T U M C O M P U T E R S Quantum computers can factorize large numbers in far fewer steps than classical computers. The best known classical algorithm requires ~ exp[1.9(log N)⅓ (log(log N))⅔ ] steps, while a quantum computer can factorize the number in ~ (log N)3 steps. In this app, you can compare the performance of a classical computer with that of a quantum computer. To make the comparison fair, we assume that both computers run at a clock speed of 2 GHz, comparable to a modern laptop.

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F A C T O R I Z I N G L A R G E N U M B E R S Factorizing is the opposite of multiplying: given a number N you have to find the prime numbers that multiply to yield N. For example: 21 = 7 x 3. While multiplying can be done very efficiently (that is, in relatively few steps), factorizing numbers is computationally very demanding. We care about factorizing a great deal in practice. The asymmetry in the level of difficulty can be exploited to create cryptographic codes in which encryption (via multiplication) is fast, but decryption without a key is slow (factorization). The cryptography used today is pretty much all based on this mathematical property.

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28588099226195682990900291063966784550769998338376898056121328236690061192846044126431325795878535247546592361778722543389224454018090665566785092906628886153218040938258860655633752601880423092053379325410243242719984296873512436640770833848829271

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Use the slider below to change the number that the computers will factorise. A larger number results in a longer computation!

80893859986434176710384245825514102012045141664667922382077099506869256194641495176663061070637918224025801788051733961312703834959504579867445593346933916066622102619619080936200720026042236156761183

2927490919440049271638806606752610852588009350078902777302235991646851254612083161454743151923178003675618091015293505795456413553675216008045771947938810907176666663822094640527257894116369146395211317951164998453192909003401405383

When you have selected the number, press the Run Test button to see how long each computer will take to factorise the number.

323497840970935236881929354804476011709302419318352968655042163072898721063142454169266923432273256244167202181295950895797919027417018189674265208781732760700896913824101396137105816246407427

When you have finished, press the Reset Test button below and try again using a different large number.

6751940008528135818416049863442945876859578795684740565417696414703747371059011276178708141561502304695703874599772523572919639580991125420634627204670186744792552072810669861771214121

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CHOOSE A NUMBER TO FACTORIZE INTO PRIMES:

8169572956241427936629055031055270797769079805278450030968765887230707053583750682595498505260037697166126582405759770658104128467889136782987595893025138458271

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5727623493074683

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739039828327034381567299677058220319189638554708896091087832019439327126869035714416584631065632711525668902901378904229192770964370825198605993333035633174856362440911910945739408642983936028158715128363076382794783350836449799519532217036144038733268117123692019148357187953079012876971586324619768966244249164438537386003347273681689605001972825810014552759

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400 ns

5 x 10–8 seconds: hydrogen fusion reaction 50 ns is roughly the time it takes for the fusion of two deuterium nuclei to fuse into helium in a fusion reactor or hydrogen bomb.

4 x 10–7 seconds: nuclear fission This is about the time between the emission of a neutron from one uranium nucleus and absorption by another nucleus.

FACTORIZING...

50 ns

5 x 10–2 seconds: woodpecker hammers A woodpecker hits the tree trunk with its beak once every 50 ms as it hollows out the tree for its nest. They can be found almost everywhere on Earth, except Australasia, Antacrtica and Madagascar.

50 ms

2 x 1012 seconds: humans leave Africa Homo sapiens evolved in Africa and migrated to other continents between 100 and 50 thousand years ago. Some Polynesian islands were not colonised until a few thousand years ago.

50 thousand years

3 μs

5 μs

3 x 10–6 seconds: shortening of the day The 2004 earth quake that shook the Indian ocean, causing the Boxing Day tsunami, reduced the length of a day on Earth by about 3 μs by changing the Earth’s moment of inertia.

5 x 10–6 seconds: strobe light Commercial high-speed strobe light produces flashes of a few microseconds, allowing for high-speed photography that freezes bullets in mid-air.

5 x 10–4 seconds: cracked glass Cracks in glass propagate at a tremendous speed. A typical crack extends over a metre in just half a millisecond.

6 x 101 seconds: put the kettle on It takes about a minute to bring half a litre of water from room temperature to boiling point in a modern kettle.

0.5 ms

1 min

1 x 10–4 seconds: fastest shutter speed A good modern digital camera can achieve shutter speeds of up to 1/8000, or about a thenth of a millisecond.

9 sec

9 x 100 seconds: 100 m world record It took Usain Bolt 9.58 seconds to run the 100 m, establishing the world record in Berlin in 2009.

0.1 ms

0.3 sec

3 x 10–1 seconds: the blink of an eye We think the blink of an eye is a short time, but it still takes 0.3 seconds to complete. It takes a classical computer this long to factor a 20-digit number, but a quantum computer can factor a 300-digit number.

0.1 ms

1 x 10–4 seconds: photon roundtrip time The Large Hadron Collider accelerates protons to nearly the speed of light (or an energy of 7 TeV) around its 27 km ring. One roundtrip of a proton is 0.1 ms.

9 x 10–5 seconds: high-frequency trading High-frequency traders buy and sell stock more than ten thousand times a second. This can sometimes cause a runaway effect, like the Flash Crash in 2010.

36 ms

4 x 10–2 seconds: hummingbird Hummingbirds hover over flowers by flapping their wings around 30 times per second. Some species flap their wings as fast as 80 times per second.

86 μs

1 x 10–3 seconds: pulsars Pulsars a very compact astronomical objects that emit a highliy focussed beam of radiation. They can rotate very fast, and produce a flash every few milliseconds as the beam sweeps across Earth.

2 x 10–5 seconds: tidal lengthening of the day The moon causes tidal forces on the Earth, causing the tides. The energy for these tides is taken from the rotation energy of the Earth, lengthening each day by this amount.

1 ms

17 μs

3 x 108 seconds: building the Suez canal The Suez canal was built from April 1859 to November 1869. It is 193 km long, and was the focal point of a political crisis between Egypt and the United Kingdom in 1956.

17 ms

10 years

2 x 10–2 seconds: see an image TVs have a refresh rate of 60Hz, which means that each image lasts about 17 ms before it is overwritten by a new one. Our brains are not fast enough to track this, and instead perceive a continuous motion.

13 ms

1 x 10–2 seconds: see an image The shortest time for the eye to see an image is 13 ms, and the brain retains the image for eight times longer in order to process its meaning.

2 x 108 seconds: light from nearby stars The nearest stars are a few lightyears away. Alpha Centauri is 4.3 light years away, and Sirius is 8.6 light years away.

5 years

10 ms

5 x 107 seconds: Oort cloud The Oort cloud is a large cloud of comets surrounding the Solar system. It starts about 60 light days from the Sun and extends to about 2 light years out.

1 x 10–2 seconds: Phelps wins The time between Michael Phelps and Milorad Cavic finishing the 100 m butterfly final in London 2012 was only 0.01 seconds.

1.5 years

9 ms

6 months

9 x 10–3 seconds: hard drive seek time On a computer hard disk that spins with 5 000 rpm, the time it takes to look up a sector is about 9 ms. By comparison, a Solid State Drive takes only 64 μs.

1 x 107 seconds: baboon gestation period Baby baboons grow in the womb for six months, as opposed to nine months for humans.

4 x 106 seconds: a human foetus At this age a human foetus is about the size of a chick pea, and has begun developing fingers and toes that are still slightly webbed at this stage.

7 weeks

7 x 10–3 seconds: spark plug A spark plug creates the ignition in a combustion engine, and the spark itself lasts a couple of milliseconds. Efficient engines should have as short and hot a spark as possible.

0.7 ms

2 weeks

1 x 106 seconds: lifetime of a mosquito Adult mosquitos live can for up to anything between a week and a few months.

5 x 10–3 seconds: conversation The time it takes to travel from your mouth to your conversaton partner’s ear is a couple of milliseconds.

5 ms

3 x 105 seconds: survive without water Water is essential for our survival. We can only go without it for about three days. Keep hydrated while your computer tries to factor this 90 digit number.

4 ms

4 x 10–3 seconds: honey bee wing flap A honey bee needs to flap its tiny wings very fast to maintain airborne: 250 times a second. They not only use them to fly, but they also use their wings to keep the hive warm at around 35 C.

3 days

16 hours

3 x 10–3 seconds: neuron It takes a neuron in the brain about 3 ms to fire. This means that shorter processes than this cannot possibly be perceived as anything other than instantaneous to the human brain.

3 ms

6 x 104 seconds: ironman It typically takes around 16 hours to complete the ironman, in which the contestant swims 2.4 miles, cycles 112 miles, and runs 26.2 miles.

2 x 10–3 seconds: breaking glass If a loud tone is resonant with the vibration frequency of a glass, the sound can make the glass shatter. This occurs typically for kHz sounds, in which each vibration lasts about two milliseconds.

1 x 104 seconds: slow-roasted pork belly The secret to a tender pork belly is to roast it slowly for a few hours at 120 C. Low temperatures allow the chewy collagen in the meat to melt into gelatine, which leads to moist and tender meat.

4 hours

2 ms

3 x 103 seconds: album of music A typical music album last a little under an hour. This is historically determined by the fact that each side of a vinyl LP can only contain 20 to 25 minutes of music.

39 min

1 x 10–3 seconds: light travels 357 km In just over 1 ms, light travels from London to Amsterdam if it would follow the curvature of the Earth. In an optical fibre, the speed of light is two thirds of that in air, so it would take a light pulse about 1.4 ms.

8 min

0.9 ms

5 x 102 seconds: light from the Sun It takes light about 8.5 minutes to travel from the Sun to the Earth. By comparison, it takes light about one million years to travel from the centre of the Sun to the surface.

9 x 10–4 seconds: Blackbird SR-17 The Blackbird SR-17 could fly at a top speed over Mach 3, which means that in 0.9 ms it would fly a distance of 1 m. When the Russians developed Mach 4.5 missiles, the Blackbird was retired.

34 thousand years

1 x 1012 seconds: age of cave paintings The first surviving human art comes in the form of cave paintings depicting hunting scenes in Chauvet cave.

5 x 10–2 seconds: Snapping your fingers creates a loud sound by hitting the palm of your hand with your middle finger at high speed.

4 x 10–2 seconds: light from LA to Sydney If light would follow the surface of the Earth, it would take 40 ms to travel from Los Angeles to Sydney.

15 thousand years

40 ms

4 x 1011 seconds: time between ice ages During an ice age average temperatures are a few degrees lower than today. Ice ages recur roughly every ten thousand years.

2 x 1011 seconds: Stonehenge The famous Wiltshire prehistoric monument is estimated to be between 4000 and 5000 years old.

4 x 10–2 seconds: a soap bubble pops A soap bubble seems to pop instantaneously, but it still takes a large bubble tens of milliseconds.

5 millennia

30 ms

3 x 10–2 seconds: washing machine When a washing machine expells water from cloths, it does so via the centrifugal force by spinning up to 1800 rpm. Each revolution takes about 30 ms.

7 x 1010 seconds: Christianity founded Two thousand years have past since the crucifiction of Jesus and the birth of Christianity. There are more than two billion christians on Earth, compared to 1.6 billion muslims.

2 millennia

3 x 1010 seconds: age of oldest universities In the early twelfth century (and late tenth century), universities in Europe were founded. Bologna dates from 1088 and Oxford from 1096.

3 x 10–2 seconds: lightning Lightning consists of a downward streamer and an upwards discharge that produces the bright flash. The total lightening strike lasts about 30 ms.

920 years

2 x 10–2 seconds: transatlantic fibre Sending a message over a transatlantic fibre from London to New York takes about 20 ms. Real connection times are much longer due to the signal processing in various routers along the way.

1 x 1010 seconds: Newton’s Principia In 1687, 328 years ago, Newton published his magnum opus, the Principia Mathematica, in which he explains the foundations of mechanics.

328 years

20 ms

100 years

2 x 10–2 seconds: atomic fire ball While the mushroom cloud is the well-known image of an atomic explosion, the interesting physics happens around 20 ms, when you can see the fire ball. The spikes are the metal support structure turning into plasma.

3 x 109 seconds: plastic bottle When left to decompose, plastic bottles take about a century to fully degrade. Before that, the plastic will have made its way into the ecosystem.

2 x 10–2 seconds: tennis ball on racket It takes time to reverse the direction of the ball on a racket. The shorter the turnaround time, the greater the force, which can be more than 200 N when returning a fast serve.

1 x 109 seconds: quantum information In 1970—45 years ago—Stephen Wiesner proposed the first application of what we now know as quantum information: quantum money that cannot be falsified. This research ultimately led to quantum computers.

45 years

6 x 1012 seconds: supervolcanic eruption By this time in the future, Earth will most likely have experienced a super-volcanic eruption that expels more than 400 cubic kilometres of magma. This will probably halt your calculation.

187 thousand years

5 x 10–2 seconds: lowest audible tone 20 Hz is on average the lowest tone humans can hear, although lower tones can be felt in the body when they are sufficiently loud.

9 x 10–2 seconds: sneeze During sneezing, you eject a large amount of mucus from your nasal cavity with force. The duration of the spray is about 90 ms.

90 ms

1 x 1014 seconds: local galactic group The radius of the volume in space containing the Local Galactic Group is 5 million light years. This is the group of galaxies around the Milky Way that are gravitationally bound to each other.

5 million years

80 ms

7 x 1013 seconds: light from Andromeda The Andromeda galaxy is two million light years away, which means it takes light two million years to get from Andromeda to us.

2 million years

8 x 10–2 seconds: airbag inflates It takes a little under 2 ms for the triggering mechanism in an airbag to initiate the inflation of the bag, which takes place at a much slower rate of 60 to 80 ms.

1 million years

7 x 10–2 seconds: heartbeat of a pygmee shrew This nervous little fellow has a heart that beats 14 times a second. Its body is about 5 cm long, and it lives in the undergrowth and eats insects and invertebrates.

70 ms

3 x 1013 seconds: use of fire by hominids One of the earliest technologies was the use of fire. Early hominids seem to have mastered this one million years ago.

436 thousand years

60 ms

1 x 1013 seconds: homo sapiens As a species, we are about half a million years old. Our closest living relative is the bonobo, an endangered type of chimpanzee living in the Congo Basin.

5 x 10–2 seconds: Ferrari 458 gear change It takes a Ferrari 458 60 ms to change gears. In normal cars with automatic transmission this takes about 200 ms, but in a Bugatti Veyron it takes only 8 ms.

2 x 1018 seconds: tidal lock Provided the Earth and the moon are not engulfed by the Sun when it turns into a red giant, by this time they will be tidally locked, and permanently facing each other.

63 billion years

0.6 s

6 x 10–1 seconds: from thought to sound The time it takes from forming a word in your mind to saying it out loud is about 600 ms.

1 x 1018 seconds: galactic pile-up The Andromeda galaxy and the Milky Way are on a galactic collision course. In roughly 25 billion years they will collide. The collision itself will play out over several billion years.

25 billion years

0.5 s

5 x 10–1 seconds: GPS roundtrip time GPS satellites are “parked” in geostationary orbits 36 thousand km above Earth, which means that they complete one orbit in 24 hours, hovering over the same spot on Earth. This leads to a long communication time.

5 x 10–1 seconds: a crickets’ chirp Crickets make a characteristic chirping sound at 5 kHz. In addition, how many times a cricket chirps each second is closely related to the temperature, with 115 chirps a minute equal to 20 C and 150 chirps 25 C.

5 x 1017 seconds: age of the universe The universe is 13.8 billion years old, and started with the Big Bang. The first galaxies formed as early as 200 million years after the Big Bang.

14 billion years

4 x 10–1 seconds: a drop of one metre It takes a object 0.44 s to fall 1 m on Earth if we ignore air drag, no matter how heavy the object is.

9.5 billion years

0.4 s

3 x 1017 seconds: lifetime of the Sun Stars the size of the Sun have a typical lifetime of about ten billion years before they go nova and become first a red giant, and then a white dwarf.

2 x 1017 seconds: formation of the Earth Nearly five billion years ago the Earth formed from the debris orbiting the Sun. Early on, a Mars-sized object imacted the Earth, creating the moon.

4 x 10–1 seconds: multiple images Modern digital SLR cameras cantypically take two and a half images per second at full resolution. However, the SD card usually cannot keep up with this for long.

5 billion years

2 billion years

4 x 10–1 seconds: Doherty threshold A computer user interface feels responsive if the time between the user action and the computer response is less than 0.4 s. This is called the Doherty threshold.

8 x 1016 seconds: oxygen The Earth’s atmosphere becomes oxygenated by single-cell organisms. This paves the way for multi-cellular organisms.

4 x 1016 seconds: multi-cellular life A billion years after life develloped on Earth, multi-cellular life evolved into a few different ways: animals, fungi, land plants, and three different types of algae.

1.2 billion years

0.3 s

3 x 10–1 seconds: jumping spider Jumping spiders creep up on their prey and jump the final distance. This takes about 0.3 s.

2 x 1016 seconds: sexual reproduction Shortly after the first multi-cellular lifeforms appeared, sexual reproduction evolved as a means of procreation. So the metabolism (the chicken) came first.

1 billion years

3 x 10–1 seconds: turning on a light bulb Switching on an incandescent light bulb is a relatively slow process. The bulb reaches maximum intensity 0.3 s after turning it on.

3 x 10–1 seconds: population growth Every 0.3 s a baby is born somewhere on this planet. The death rate is one person every 0.5 s, which means that the global population is still growing fast.

1 x 1016 seconds: first land vertebrates Around 350 million years ago the first tetrapods crawled from the water onto land, starting the colonisation of the continents by animals.

346 million years

0.2 s

6 x 1015 seconds: flowers 150 million years ago the first flowers appear, along with pollenating insects.

140 million years

2 x 10–1 seconds: acceptable gaming latency One of the most important figures of merit in online gaming is the ping time, or the latency. Slow-paced games can deal with 0.2 s of latency, but fast online shooters require below 50 ms.

2 x 10–1 seconds: human reaction time It takes about 0.2 seconds from the time you touch a hot stove to the involuntary retraction of your hand.

3 x 1015 seconds: dinosaurs The T-rex and his friends roam the Earth, and birds begin to replace the pterosaurs. It’s still 24 million years until the meteor that hit Chicxulub and wiped out the dinosaurs.

89 million years

0.1 s

1 x 1015 seconds: grass The first grasses appear about 40 million years ago, well after the dinosaurs have gone extinct.

42 million years

1 x 10–1 seconds: the black mamba The black mamba is one of the fastest (and deadliest) snakes on the planet. It can move at a speed of 25 km/h, which means that it travels 1 m in only 0.14 seconds.

6 x 1014 seconds: glass It takes glass about 20 million years to degrade, and this makes it one of the most durable materials known to man.

20 million years

1 x 10–3 seconds: engine stroke When idling, a car’s engine is still running. Each stroke of the engine last about a tenth of a second. When accelerating, the engine stroke can become ten times as fast.

3 x 1014 seconds: the first apes Our ancestors—the apes—evolved about ten million years ago. Primates tend to live in trees in (sub-) tropical regions. Humans are the only apes that migrated to all continents.

10 million years

1 x 10–1 seconds: false start In the Olympics, if the athlete starts running within a tenth of a second after the starting shot, it is considered a false start.

1 s

4 x 1019 seconds: star formation stops Due to the lack of available interstellar dust in nebulae, star formation will come to a halt. This means that the lights in the universe are going out. For the rest of time the universe will be dark.

1200 billion years

1 x 100 seconds: the second The second is our unit of time, defined in the Système International as the duration of 9 192 631 770 periods of the radiation of two hyperfine levels of a caesium atom.

2 x 1019 seconds: black hole evaporation Primordial black holes of 1011 kg (the size of a small asteroid) take this long to evaporate via Hawking radiation. Stellar black holes take much, much longer.

680 billion years

0.9 s

9 x 10–1 seconds: heart beat The average heart beat rate of an adult human at rest is about 70 beats per minute, which translates to one beat every 0.9 seconds.

0.8 s

1 x 1019 seconds: Local Group merger Around this time the 47 galaxies of the Local Group will have merged into one large supergalaxy.

370 billion years

8 x 10–1 seconds: sound travels 280 m If you shout from the top of the Shard in London it takes the sound 0.8 seconds to reach street level, still well in time to warn passers by about the falling flower pot.

7 x 1018 seconds: big bang echo fades away The cosmic microwave background radiation—currently at 2.7 K—cools further to 0.3 K. This is undetectable with current technology.

200 billion years

8 x 10–1 seconds: burning a match head The head of a match will keep going in strong winds (or even under water) because the head has its own oxygen supply. It burns out in roughly 0.8 s.

7 x 10–1 seconds: halflife of C16 Carbon comes in a variety of isotopes. A nucleas with 6 protons and 10 neutrons is unstable, however, and such a nucleaus has a 50:50 chance of decaying after 0.7 s.

0.7 s

4 x 1018 seconds: cosmic isolation The universe’s expansion causes all galaxies beyond the Milky Way’s Local Group to disappear beyond the cosmic horizon, removing them from the observable universe.

114 billion years