Quantum Computers and Brahmagupta — yes the two go together!

Gabriellebirchak/ July 23, 2020/ Ancient History, Classical Antiquity, Uncategorized

Podcast Transcript and Blog

I’m going to start my pod­cast with a digres­sion because today I read excit­ing news. The Nation­al sci­ence foun­da­tion has award­ed the Uni­ver­si­ty of Cal­i­for­nia Berke­ley $25 mil­lion to cre­ate a mul­ti-uni­ver­si­ty insti­tute that will advance quan­tum sci­ence to cre­ate quan­tum com­put­ers. This news excites me because it’s going to take math to a whole new lev­el and make it pow­er­ful beyond num­bers and words. Though quan­tum com­put­ing is young, the oppor­tu­ni­ties that it presents are tru­ly mind-blowing.

Unlike the con­ven­tion­al com­put­ers that we use today, where infor­ma­tion is stored as bits, quan­tum com­put­ers store infor­ma­tion and data as qubits, also known as quan­tum bits. You see, the com­put­er that you use today is stream­ing elec­tri­cal puls­es that rep­re­sent ones or zeros. Though quan­tum com­put­ers also use zeros and ones, they also use super­po­si­tion and entan­gle­ment to cre­ate a third state of qubits that rep­re­sent a zero and a one simultaneously.

We do not have quan­tum com­put­ers for pub­lic use yet because we still are exe­cut­ing com­pu­ta­tion­al oper­a­tions on a small num­ber of qubits. Fur­ther­more, these process­es require a com­put­er to oper­ate at neg­a­tive 460 degrees Fahren­heit, also known as neg­a­tive 273 degrees Cel­sius, also known as zero Kelvin. 

Quan­tum com­put­ers are going to be the future of com­put­ing. Used today, they can solve prob­lems that would take a con­ven­tion­al com­put­er bil­lions of years to solve because they are thou­sands of times faster. Because of their pow­er, they will be able to break encryp­tions. How­ev­er, also because of their pow­er, they will be able to cre­ate unhack­able sys­tems. Also, instead of using elec­tric­i­ty, as our con­ven­tion­al com­put­ers do, quan­tum com­put­ers use quan­tum tun­nel­ing for pow­er. As a result, quan­tum com­put­ers will reduce the con­sump­tion of pow­er that a com­put­er gen­er­ates. But, to me, the most excit­ing thing that a quan­tum com­put­er can do is find a prod­uct rep­re­sen­ta­tion for the solu­tion of Pell’s equa­tion in poly­no­mi­al time. Poly­no­mi­al-time is the time it takes to do a com­pu­ta­tion, depend­ing on the amount of inputted data. That is actu­al­ly irrel­e­vant to this pod­cast, but it’s good to know for inter­est­ing par­ty conversation.

What is rel­e­vant is that a quan­tum com­put­er can find a prod­uct rep­re­sen­ta­tion for the solu­tion of Pell’s equa­tion. You see, Pell’s equa­tion, also known as the Pell-Fer­mat equa­tion is any Dio­phan­tine equa­tion of the form

x^2 -ny^2 =1

Where the vari­able n rep­re­sents a pos­i­tive non­square inte­ger. In this equa­tion, solu­tions are solved for x and y. When it’s pre­sent­ed in Carte­sian coor­di­nates and n is pos­i­tive, it is shown as a hyperbola. 

When n is neg­a­tive, it is an ellipse.

This par­tic­u­lar equa­tion, accord­ing to Joseph Lagrange, has infi­nite inte­ger solu­tions as long as the vari­able n is not a per­fect square, and x and y are integers.

Though Dio­phan­tine equa­tions have been around since the math­e­mati­cian Dio­phan­tus pre­sent­ed his work called Arith­meti­ca around 250 CE. How­ev­er, it wasn’t until 628 when Brah­magup­ta dis­cov­ered an inte­ger solu­tion to this equa­tion where n=92.

Brah­magup­ta was a tremen­dous math­e­mati­cian. The world-renowned sci­ence his­to­ri­an, George Sar­ton, stat­ed that Brah­magup­ta was “one of the great­est sci­en­tists of his race and the great­est of all time.”

Brah­magup­ta was born in 598 CE in India. In 628, he wrote and improved Trea­tise of Brah­ma called Brah­mas­phutasid­dhān­ta. The book con­sists of 24 chap­ters and has 1008 vers­es. It includes astron­o­my, alge­bra, geom­e­try, trigonom­e­try, and algo­rith­mics. The solu­tion to the ref­er­enced Dio­phan­tine equa­tion is found in chap­ter 18 of Brah­mas­phutasid­dhān­ta. Though it seems like a sim­ple process today, he was solv­ing for large num­bers using Diophantus’s process of syn­co­pat­ed alge­bra, which was the process of using sim­ple sym­bol­ism before it evolved into sym­bol­ic alge­bra. As a result, Brahmagupta’s time, he only had sym­bols that rep­re­sent­ed expo­nents, sub­trac­tion, and an equal sign. Sym­bol­ic alge­bra wouldn’t be used until the six­teenth century.

The con­tri­bu­tion of Brahmagupta’s work to math­e­mat­ics is exten­sive. For exam­ple, in arith­metic, we have four fun­da­men­tal oper­a­tions, which are addi­tion, sub­trac­tion, mul­ti­pli­ca­tion, and divi­sion. Even though these fun­da­men­tal oper­a­tions had been around for cen­turies, the cur­rent sys­tem that we know is based on the Hin­du Ara­bic num­ber sys­tem, and it appeared in Brah­mas­phutasid­dhān­ta.

And speak­ing of zeros and ones, Brah­mas­phutasid­dhān­ta was the first exist­ing book that treat­ed zero as a num­ber and not just a place­hold­er. Before this, in Ptole­ma­ic math­e­mat­ics, zero was only used as a place­hold­er. How­ev­er, though Brah­magup­ta is con­tributed to treat­ing zero as a num­ber, in 2017, research uncov­ered an Indi­an man­u­script the dates as far back as 200 CE. It showed the use of zero as a num­ber. Regard­less, we would not have been intro­duced to this con­cept if it wasn’t for the work that Brah­magup­ta pre­sent­ed in Brah­mas­phutasid­dhān­ta.

Brah­magup­ta was also one of the first to pro­vide rules for adding and mul­ti­ply­ing neg­a­tive numbers. 

All of this played into sev­er­al rules that showed that the sum of two pos­i­tives are pos­i­tive:

1+1=2

The sum of two neg­a­tives are negative:

(-1)+(-1)=-2

The sum of a pos­i­tive and a neg­a­tive is the difference:

(-1)+3=2

And if the sum of a pos­i­tive and a neg­a­tive are equal, the answer is zero:

(-3)+3=0

Brah­magup­ta also described the process of zero and neg­a­tive num­bers in mul­ti­pli­ca­tion. He showed that the prod­uct of a neg­a­tive and a pos­i­tive is negative:

2 \times -2=-4

The prod­uct of two neg­a­tives is positive:

-2 \times-2=4

The prod­uct of two pos­i­tives is positive:

2 \times 2=4

He also showed that the prod­uct of zero and a neg­a­tive is zero:

0 \times -2=0

And the prod­uct of zero and a pos­i­tive is also zero:

0 \times 2=0

For the sides of the right-angled tri­an­gle, he pre­sent­ed two sets of Pythagore­an triples, which were:

2mn, m^2-n^2, and\space m^2+n^2

and

\sqrt m, \space \frac{1}{2}(\frac{m}{n}-m), \space  and \space  \frac{1}{2}(\frac{m}{n}+m)

In geom­e­try, he cre­at­ed an exact for­mu­la for find­ing the area of cyclic quadri­lat­er­als. A cyclic quadri­lat­er­al is a quadri­lat­er­al that is inscribed with­in a cir­cle where all the ver­tices sit on the sin­gle cir­cle.[i]

That for­mu­la showed that the area of any quadri­lat­er­al whose sides are a, b, c, d is

Area= \sqrt{\smash[b]{(s-a)(s-b)(s-c)(s-d)}}

Where

s=\frac{1}{2}(a+b+c+d)

His work with cyclic quadri­lat­er­als led to Brahmagupta’s famous the­o­rem, which stat­ed that if a cyclic quadri­lat­er­al has per­pen­dic­u­lar diag­o­nals, then the per­pen­dic­u­lar to a side from a point of inter­sec­tion of the diag­o­nals bisects the oppo­site side.[ii]

In oth­er words, for the fol­low­ing image:

\overline{BD}\perp \overline{AC}, \space \overline{EF}\perp \overline{BC}\implies\ |\overline{AF}|=|\overline{FD}|

In astron­o­my, though Brah­magup­ta believed that the sun orbit­ed the Earth, he pre­sent­ed an argu­ment for show­ing that the moon is clos­er to the Earth than the Sun.

The evo­lu­tion of math­e­mat­ics is such a beau­ti­ful one.  Brahmagupta’s work helped to speed up the cal­cu­la­tion process of the Dio­phan­tine equa­tion. Today we begin to move for­ward into a new chap­ter of quan­tum com­put­ing, as the process of crunch­ing large num­bers and data becomes faster and eas­i­er. Though Moore’s law is becom­ing out­dat­ed, and though we have many prob­lems to cor­rect includ­ing noise and error cor­rec­tion, and though we may not see quan­tum com­put­ers for a while, and though it is high­ly like­ly that quan­tum com­put­ers may not out­pace or even replace con­ven­tion­al com­put­ers, ini­tial­ly, we are step­ping into a new chap­ter where even our fastest com­put­ers today will become anti­quat­ed. With this progress, the num­bers that we under­stand today may change by def­i­n­i­tion and rep­re­sen­ta­tion. As a result, as math­e­mati­cians and sci­en­tists, we stand at a place in time that serves as a mark­er for the future of math­e­mat­ics. Yeah! We stand along a lin­ear path, specif­i­cal­ly at that point between his­to­ry and the future, where we can see all that brought us here math­e­mat­i­cal­ly, and all that will take us to greater under­stand­ings. It sounds a bit dra­mat­ic and poetic…but it also sounds very exciting!

[i]. David E. Smith, His­to­ry of Math­e­mat­ics (North Chelms­ford: Couri­er Cor­po­ra­tion, 1958), 158.

[ii]. Michael J. Bradley, The Birth of Math­e­mat­ics: Ancient Times To 1300 (New York: Infobase Pub­lish­ing, 2006), 70.