A Brief History of Computing Quantum Computing, Communication, and Cryptography

Introduction What we learn from history? The physical layer Quantum mechanics Content of the course

A brief history of computing Quantum computing, communication, and cryptography

Dimitri Petritis

Institut de recherche mathématique de Rennes Université de Rennes 1 et CNRS (UMR 6625)

Coëtquidan, 7 January 2017

Coëtquidan, 7 January 2017 QCCC Introduction What we learn from history? The physical layer The rôle of quantum mechanics Quantum mechanics Content of the course The problem

Entire domains of scientiﬁc (and generally human) activity rely on retrieval, processing, transmission, and protection of information. Nowadays: those steps are algorithmically automated by programmes executed on reliable electronic devices. We can argue using the abstract mathematical categories of logical circuits of a computer without paying attention to the physical layer on which these programmes are executed. Because we can do so presently — and still for some short time!

Coëtquidan, 7 January 2017 QCCC Introduction What we learn from history? The physical layer The rôle of quantum mechanics Quantum mechanics Content of the course The problem

Coëtquidan, 7 January 2017 QCCC Before the . . . revolution (≤ 1946) Introduction Pre-history (' 1946) What we learn from history? Proto-history 1947–1956 The physical layer First revolution: the transistor (1956–1971) Quantum mechanics The second revolution: the microprocessor (1971–2020?) Content of the course The end of certainty (≥ 2010) What we learn from history? < 1946 Charles Babbage (London 1791– London 1871) invented a machine computing and printing values of polynomials; functioned with punch cards modelled after the automated looms of Joseph Marie Jacquard.

Figure: Charles Babbage and . . . his model: the loom of Jacquard.

Figure: The analytic machine constructed by Babbage and the punch cards used for its programming.

Coëtquidan, 7 January 2017 QCCC Before the . . . revolution (≤ 1946) Introduction Pre-history (' 1946) What we learn from history? Proto-history 1947–1956 The physical layer First revolution: the transistor (1956–1971) Quantum mechanics The second revolution: the microprocessor (1971–2020?) Content of the course The end of certainty (≥ 2010) What we learn from history? Programming Babbage’s machine Augusta Ada King, countess of Lovelace, born Ada Byron (London 1815 – London 1852) writes an “algorithm” to compute Bernoulli’s numbers (Bn) n m X 1 X S (n) = km = C k B nm+1−k . m m + 1 m+1 k k=1 k=0 First algorithm conceived to run on a Babbage’s machine.

Coëtquidan, 7 January 2017 QCCC Before the . . . revolution (≤ 1946) Introduction Pre-history (' 1946) What we learn from history? Proto-history 1947–1956 The physical layer First revolution: the transistor (1956–1971) Quantum mechanics The second revolution: the microprocessor (1971–2020?) Content of the course The end of certainty (≥ 2010) What we learn from history? The contribution of Turing Alan Mathison Turing (London 1912 – Cheshire 1954), mathematician, logician, cryptanalyst and . . . (preposterously!) computer scientist; broke the Enigma ciphering used by German sub-marines in 2nd world war with the help of the machine of his invention named “the bomb”.

Figure: Alan Turing and one of the ca. 200 replicas of “the bomb”. Coëtquidan, 7 January 2017 QCCC Before the . . . revolution (≤ 1946) Introduction Pre-history (' 1946) What we learn from history? Proto-history 1947–1956 The physical layer First revolution: the transistor (1956–1971) Quantum mechanics The second revolution: the microprocessor (1971–2020?) Content of the course The end of certainty (≥ 2010) What we learn from history? The contribution of von Neumann Margittai Neumann János Lajos Budapest 1903 – John von Neumann Princeton 1954, mathematician and physicist with major contributions in quantum mechanics, functional analysis, theory of sets, computer sciences (cybernetics), game theory and various other domains of physics and mathematics. Participated in the American military programmes.

Figure: John von Neumann and the schematic view of a computer architecture named after him. Coëtquidan, 7 January 2017 QCCC Before the . . . revolution (≤ 1946) Introduction Pre-history (' 1946) What we learn from history? Proto-history 1947–1956 The physical layer First revolution: the transistor (1956–1971) Quantum mechanics The second revolution: the microprocessor (1971–2020?) Content of the course The end of certainty (≥ 2010) What we learn from history? ca. 1946

By that time we had to care about the physical layer! 1946: Electronic numerical integrator and computer (ENIAC) ﬁrst universal computer contracted by the engineer John Adam Presper Eckert Jr. (Philadelphia, PA, 1919 – Bryn Mawr, PA, 1995) and the physicist John William Mauchly (Cincinatti, OH, 1907 – Ambler, PA, 1980). 19000 tubes mass: 30 tonnes footprint: 72m2 electric power: 140kW clock frequency: 100kHz (330 multiplications per second).

Figure: The machine room of ENIAC.

Figure: A technician changing one of the 19000 tubes of ENIAC.

Figure: Two operators in the process of . . . programming ENIAC.

Figure: The monster can now compute.

Coëtquidan, 7 January 2017 QCCC Before the . . . revolution (≤ 1946) Introduction Pre-history (' 1946) What we learn from history? Proto-history 1947–1956 The physical layer First revolution: the transistor (1956–1971) Quantum mechanics The second revolution: the microprocessor (1971–2020?) Content of the course The end of certainty (≥ 2010) What we learn from history? 1947 Décembre 1947: Invention of the transistor by three physicists William Bradford Shockley (London 1910 – Palo Alto, CA 1989), Walter Houser Brattain (Amoy, China, 1902 – Seattle, WA, 1987) et John Bardeen (Madison, WI, 1908 – Boston, MA, 1991) of Bell Telephone Laboratories.

Figure: Bardeen, Brattain et Shockley and a . . . failed welding that became the transistor!

January 1948 : Wallace Eckert from IBM and his team completed the SSEC (Selective Sequence Electronic Calculator). June 1948 : NewMan, William and their team from the university of Manchester completed a prototype called Manchester Mark I. August 1949: P. Eckert et J. Mauchly (having created their own company) develop the ﬁrst bi-processor computer: the BINAC for the US Navy. The two processors execute the same operations in parallel to increase the computing reliability. 1950: The computer developed during the war by Konrad Zuse, the Z4, is ﬁnally assembled at the ETHZ; its programming allows jumps and conditional branchings.

1950: Assembler, invented par Maurice V. Wilkes of the university of Cambridge, replaced the binary. January 1951: Construction of the ﬁrst Soviet computer MESM under the leadership of Sergei Alexeevich Lebedev at the Academy of Sciences of Ukraine. 1951: Magnetic drum mass storage (depicted a model of 10 kbit capacity!)

installed on ERA 1101 of 1 Mbit. 1951: P. Eckert and J. Mauchly start selling the UNIVAC I (UNIversal Automatic Computer). First commercial computer performing 8333 additions or 555 multiplications per second. 56 units sold. List price: Computer: 750 000 USD (1951) = 7 095 750 USD (2016), Fast printer: 185 000 USD (1951) = 1 750 285 USD (2016).

1952: IBM produces the IBM 701 for the Defence (19 units sold). Tube memory of 2048 or 4096 words of 36 bits; 16000 additions or 2200 multiplications per second. The first unit installed at Los Alamos served for the thermo-nuclear bomb project. 1952: First French computer, CUBA (Calculateur Universel Binaire de l’Armement), produced by the company SEA. July 1953: IBM first commercial computer produced in series: the IBM 650, conceived to be compatible with the electromechanical accounting machines with punched cards. 1955: First commercial computer network: SABRE (Semi Automated Business Related Environment) developed by IBM, connects 1200 telex machines through US for flight reservations of American Airlines. 1955: IBM 704 developed by Gene Amdahl. First commercial machine with a mathematic coprocessor. 5 kflops (thousand of floating point operations per second). Very reliable machine; out of order . . . only once per week on average.

1956: First computer containing transistors, constructed by Bell Company, the TRADIC, heralding the second generation of computers. 1958: First commercial entirely transistorised computer, the CDC 1604, developed by Seymour Cray. 1959: Demonstration of the ﬁrst integrated circuit by Texas Instruments. 1961: MAC (Multi Access Computer) project of MIT directed par John Mc Carthy. Goal: allow multi-users sharing of the resource. 1970: First memory chip by Intel equivalent of 1024 very bulky ferrite memories on 0.5 × 0.5mm2 squares (capacity: 1kbit, i.e. 128 octets).

Novembre 1971: Intel sells the ﬁrst microprocessor Intel 4004, conceived by Marcian Hoﬀ. Processor 4 bits at 108 KHz, Addresses 640 octets of memory, 60000 instructions per second, 2300 transistors in 10 µm technology, Price: 200 $.

Figure: Gordon Earl Moore (San Francisco, CA, 1929 –), the marine’s oﬃcer who proposed the so called “Moore’s law” (in the middle) and one of most recent microprocessors (Intel Core I7 Nehalem (2008), with a surface of 263 mm2).

Processor Transistor count Year Designer Process Area Pentium 4 Prescott-2M 169 000 000 2005 Intel 90 nm 143 mm² Intel 4004 2 300 1971 Intel 10 000 nm 12 mm² Pentium D Smithfield 228 000 000 2005 Intel 90 nm 206 mm² Intel 8008 3 500 1972 Intel 10 000 nm 14 mm² Pentium 4 Cedar Mill 184 000 000 2006 Intel 65 nm 90 mm² Motorola 6800 4 100 1974 Motorola 6 000 nm 16 mm² Pentium D Presler 362 000 000 2006 Intel 65 nm 162 mm² Intel 8080 4 500 1974 Intel 6 000 nm 20 mm² Core 2 Duo Wolfdale 411 000 000 2007 Intel 45 nm 107 mm² RCA 1802 5 000 1974 RCA 5 000 nm 27 mm² AMD K10 quad-core 2M L3 463 000 000 2007 AMD 65 nm 283 mm² MOS Technology 6502 3 510 1975 MOS Technology 8 000 nm 21 mm² POWER6 789 000 000 2007 IBM 65 nm 341 mm² Intel 8085 6 500 1976 Intel 3 000 nm 20 mm² Core 2 Duo Wolfdale 3M 230 000 000 2008 Intel 45 nm 83 mm² Zilog Z80 8 500 1976 Zilog 4 000 nm 18 mm² Core i7 (Quad) 731 000 000 2008 Intel 45 nm 263 mm² Motorola 6809 9 000 1978 Motorola 5 000 nm 21 mm² AMD K10 quad-core 6M L3 758 000 000 2008 AMD 45 nm 258 mm² Intel 8086 29 000 1978 Intel 3 000 nm 33 mm² Six-core Opteron 2400 904 000 000 2009 AMD 45 nm 346 mm² Intel 8088 29 000 1979 Intel 3 000 nm 33 mm² 16-core SPARC T3 1 000 000 000 2010 Sun/Oracle 40 nm 377 mm² Motorola 68000 68 000 1979 Motorola 3 500 nm 44 mm² Six-core Core i7 (Gulftown) 1 170 000 000 2010 Intel 32 nm 240 mm² WDC 65C02 11 500 1981 WDC 3 000 nm 6 mm² 8-core POWER7 32M L3 1 200 000 000 2010 IBM 45 nm 567 mm² Intel 80186 55 000 1982 Intel 3 000 nm 60 mm² Quad-core z196[20] 1 400 000 000 2010 IBM 45 nm 512 mm² Intel 80286 134 000 1982 Intel 1 500 nm 49 mm² Quad-core Itanium Tukwila 2 000 000 000 2010 Intel 65 nm 699 mm² WDC 65C816 22 000 1983 WDC 9 mm² 8-core Xeon Nehalem-EX 2 300 000 000 2010 Intel 45 nm 684 mm² Motorola 68020 190 000 1984 Motorola 2 000 nm 85 mm² Quad-core + GPU Core i7 1 160 000 000 2011 Intel 32 nm 216 mm² Intel 80386 275 000 1985 Intel 1 500 nm 104 mm² 10-core Xeon Westmere-EX 2 600 000 000 2011 Intel 32 nm 512 mm² MultiTitan 180 000 1988 DEC WRL 1 500 nm 61 mm² 8-core AMD Bulldozer 1 200 000 000 2012 AMD 32 nm 315 mm² Intel 80486 1 180 235 1989 Intel 1 000 nm 173 mm² Quad-core + GPU AMD Trinity 1 303 000 000 2012 AMD 32 nm 246 mm² R4000 1 350 000 1991 MIPS 1 000 nm 213 mm² Six-core zEC12 2 750 000 000 2012 IBM 32 nm 597 mm² Pentium 3 100 000 1993 Intel 800 nm 294 mm² 8-core Itanium Poulson 3 100 000 000 2012 Intel 32 nm 544 mm² SA-110 2 500 000 1995 Acorn/DEC/Apple 350 nm 50 mm² 61-core Xeon Phi 5 000 000 000 2012 Intel 22 nm 720 mm² Pentium Pro 5 500 000 1995 Intel 500 nm 307 mm² Apple A7 (dual-core ARM64) 1 000 000 000 2013 Apple 28 nm 102 mm² AMD K5 4 300 000 1996 AMD 500 nm 251 mm² Six-core Core i7 Ivy Bridge E 1 860 000 000 2013 Intel 22 nm 256 mm² Pentium II Klamath 7 500 000 1997 Intel 350 nm 195 mm² 12-core POWER8 4 200 000 000 2013 IBM 22 nm 650 mm² AMD K6 8 800 000 1997 AMD 350 nm 162 mm² Xbox One main SoC 5 000 000 000 2013 Microsoft/AMD 28 nm 363 mm² Pentium II Deschutes 7 500 000 1998 Intel 250 nm 113 mm² Quad-core Core i7 Haswell 1 400 000 000 2014 Intel 22 nm 177 mm² Pentium III Katmai 9 500 000 1999 Intel 250 nm 128 mm² 8-core Core i7 Haswell-E 2 600 000 000 2014 Intel 22 nm 355 mm² Pentium II Mobile Dixon 27 400 000 1999 Intel 180 nm 180 mm² Apple A8X (tri-core ARM64) 3 000 000 000 2014 Apple 20 nm 128 mm² Pentium III Coppermine 21 000 000 2000 Intel 180 nm 80 mm² 15-core Xeon Ivy Bridge-EX 4 310 000 000 2014 Intel 22 nm 541 mm² Pentium 4 Willamette 42 000 000 2000 Intel 180 nm 217 mm² 18-core Xeon Haswell-E5 5 560 000 000 2014 Intel 22 nm 661 mm² Pentium III Tualatin 45 000 000 2001 Intel 130 nm 81 mm² Quad-+ore i7 Skylake K 1 750 000 000 2015 Intel 14 nm 122 mm² Pentium 4 Northwood 55 000 000 2002 Intel 130 nm 145 mm² Duo-Core i7 Broadwell-U 1 900 000 000 2015 Intel 14 nm 133 mm² Itanium 2 Madison 6M 410 000 000 2003 Intel 130 nm 374 mm² IBM z13 3 990 000 000 2015 IBM 22 nm 678 mm² Pentium 4 Prescott 112 000 000 2004 Intel 90 nm 110 mm² IBM z13 Storage Controller 7 100 000 000 2015 IBM 22 nm 678 mm² Itanium 2 with 9 MB cache 592 000 000 2004 Intel 130 nm 432 mm² 22-core Xeon Broadwell-E5 7 200 000 000 2016 Intel 14 nm 456 mm²

Silicium (Si) = solid crystallising in the “diamond” shape. Extracted from raw material in abundance on Earth: sand.

Periodic repetition√ of a cube having edges of a = 0.5431 nm. Interatomic 3 distance 4 a = 0.2352 nm.

Silicium (Si) = solid crystallising in the “diamond” shape. Extracted from raw material in abundance on Earth: sand.

Periodic repetition√ of a cube having edges of a = 0.5431 nm. Interatomic 3 distance 4 a = 0.2352 nm.

For comparison: thickness of human hair 100µm, 0.032µm= 32nm (2010), 0.022µm= 22nm (2013 - source Intel), 0.014µm= 14nm (2016 - source Intel). interatomic distance of silicium 235.2pm' 0.2352nm. In 2016: etching thickness = 60 interatomic distances of silicium.

The increased reliability of the microprocessor technology prevailing in modern computers (3rd generation) made us forget that beneath the logical layer lies a physical layer. Some 10–50 Silicium atoms to build a transistor ⇒ quantum behaviour. Not a shortage of raw material but a scarcity of atoms in Silicium crystal. The exponential development leads us directly to the 4th generation of computers (quantum), where we must restart caring about the underlying physical layer, change our mental frames because quantum physics violates our classical intuition.

Physics is an experimental science

Experiment Phenomenology

Theory Model

Physics is an experimental science

Experiment Phenomenology

Theory Model

Coëtquidan, 7 January 2017 QCCC Introduction The experimental cycle What we learn from history? The probabilistic nature of scientific predictions The physical layer The infinitely small and the infinitely large Quantum mechanics Mental categories Content of the course Illustration the endless cycle of physics Example: A piston containing a perfect gas, almost isolated from the rest of the world. Precise preparation of the system: state. The experiment: Interaction with measuring apparatus manometre (measures observable p), thermometre (measures observable T ), rule (measures observable V ), inducing only negligible perturbation on the state. The phenomenology: pV /T = const (Boyle-Mariotte law). The model (proposed after many other experiments of the same nature are performed): thermodynamics of perfect gases. The theory (microscopic explanation): kinetic theory of gases. (More complete theory: statistical physics.)

System carefully prepared in some state ρ ∈ S. Some observable X ∈ O chosen to be measured. Interaction of system with apparatus designed to measure the outcomes of X . Every measurement contains errors. Nevertheless, statistical reproducibility ⇒ probability law on the space of possible outcomes of the observables. Hence, experiment = mapping ρ S × O 3 (ρ, X ) 7→ ν := νX ∈ M1(X).

Physical theories are not eternal truths: remain valid as long as no experiment contradicts them! Coëtquidan, 7 January 2017 QCCC Introduction The experimental cycle What we learn from history? The probabilistic nature of scientific predictions The physical layer The infinitely small and the infinitely large Quantum mechanics Mental categories Content of the course What are the lessons learnt from experimental nature of phenomena?

Axioms: models, theories. Theorems: phenomenology. Exist per se and forever. No need of experimental veriﬁcation. Experimental character of mathematics: hidden in the intuition of the mathematician. Mathematical physics: physics (thruth relies on experimental observation/veriﬁcation), mathematics (phenomenology must come from theorems).

Measure units introduced during French Revolution for everyday quantities to have reasonable numerical values, typically 10−3 − 103. Length l: 10−15m (proton radius) – 1026m (radius of the universe). Mass m: 10−30kg (electron mass) – 1050kg (mass of the universe). Time t: 10−23s (time for light to cross atomic nucleus) – 1017s (age of the universe).

Two physical constants: −34 Planck’s constant ~ = 10 Js, (mosquito 2.5mg, 1cm/s produces action −10 24 2.5 × 10 Js= 2.5 × 10 ~), speed of light in vacuum c = 3 × 108m/s, (Mach 1= 343m/s= 1.1 × 10−6c).

Quantum ﬁeld theory ~ → 0 c → ∞

Special relativity Quantum mechanics c → ∞ ~ → 0

Classical mechanics

Coëtquidan, 7 January 2017 QCCC Introduction What we learn from history? The physical layer Diﬀerences with classical mechanics Quantum mechanics Content of the course Probabilistic nature

Statistical reproducibility of experimental results: many repetitions ⇒ smaller and smaller random fluctuations. Large classes of phenomena (e.g. celestial motion) exist where random fluctuations negligible. Determinism of 18th and 19th centuries. Large classes of phenomena exist (e.g. heads or tails) where random fluctuations are important. Deterministic description of the motion but random initial condition. There exist large classes of phenomena (e.g. behaviour of small atomic or subatomic particles) where random fluctuations are important. Intrinsically stochastic description, irreducible to classical approch.

Coëtquidan, 7 January 2017 QCCC Introduction What we learn from history? The physical layer Diﬀerences with classical mechanics Quantum mechanics Content of the course Probabilistic nature

Coëtquidan, 7 January 2017 QCCC Introduction What we learn from history? Can we ignore QM? The physical layer Failures of RSA Quantum mechanics Programme of the course Content of the course Interest

Theoretical interest: not a single prediction of quantum mechanics has been falsiﬁed by experiment. Mathematical interest: algebra, analysis, probability and statistics (generalised into a non-commutative framework). Interest for other fundamental sciences: atomic and molecular physics hence chemistry hence biology. Technological interest (1/3 of world’s economy). Several existing industrial applications (tunnel eﬀect microscopy, lasers, quantum cryptography and communication) or foreseen (quantum computing).

Coëtquidan, 7 January 2017 QCCC Introduction What we learn from history? Can we ignore QM? The physical layer Failures of RSA Quantum mechanics Programme of the course Content of the course Interest

Coëtquidan, 7 January 2017 QCCC Introduction What we learn from history? Can we ignore QM? The physical layer Failures of RSA Quantum mechanics Programme of the course Content of the course Can we ignore Quantum Mechanics?

NO!

p et q large primes, N = pq, n = log N Beginnings of RSA protocol (1978), τ = O(exp(n)). Lenstra-Lenstra (1997), τ = O(exp(n1/3(log n)2/3)). Shor (1994), if a quantum computer existed τ = O(n3). Very rough estimation: 1 operation par nanosecond, n = 1000

O(exp(n)) O(exp(n1/3(log n)2/3)) O(n3) 10417 a 1 0.2 a 1 s

1For comparison: age of the universe 1.5 × 1010 a. Coëtquidan, 7 January 2017 QCCC Introduction What we learn from history? Can we ignore QM? The physical layer Failures of RSA Quantum mechanics Programme of the course Content of the course Factoring large integers ...... having only two prime factors

O(exp(n)) O(exp(n1/3(log n)2/3)) O(n3) 10417 a 1 0.2 a 1 s

YES!

At pre-industrial level: quantum communication, and quantum cryptography. In foreseen future: quantum computing.

Some reminders of classical computing. Some reminders of probability theory. Some reminders of Hilbert spaces with emphasis on tensor products. Postulates of quantum mechanics viewed as a non-commutative version of probability theory. Quantum gates, quantum circuits. Principles of quantum computing. Shor’s algorithm for factoring into primes. Other quantum algorithms. Error correcting codes. Introduction to quantum communication and cryptography.

Coëtquidan, 7 January 2017 QCCC