The Logic of Quantum Computing
Marco Vitturi The Logic of
Timeline of Quantum Computing QC
Qubits Information Marco Vitturi Reversibility
Query School of Mathematics complexity
Probability vs. Quantum Joint PG Colloquium - 30 April 2014 mechanics
Simon’s algorithm
Factoring
The logic of quantum computing Timeline of QC
The Logic of 1970 2014 Quantum Computing
Marco Vitturi 1973 Timeline of Holevo QC studies Qubits information in Information qubits
Reversibility
Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing Timeline of QC
The Logic of 1970 2014 Quantum Computing
Marco Vitturi 1973 Timeline of Holevo QC studies Qubits information in Information qubits
Reversibility Query 1981 complexity Feynman calls for Probability vs. Quantum a computer mechanics based on QM for
Simon’s simulation algorithm purposes
Factoring
The logic of quantum computing Timeline of QC
The Logic of 1970 2014 Quantum Computing
Marco Vitturi 1973 1985 Timeline of Holevo Deutsch QC studies introduces the Qubits information in Universal Information qubits Quantum Turing Machine Reversibility Query 1981 complexity Feynman calls for Probability vs. Quantum a computer mechanics based on QM for
Simon’s simulation algorithm purposes
Factoring
The logic of quantum computing Timeline of QC
The Logic of 1970 2014 Quantum Computing
Marco Vitturi 1973 1985 Timeline of Holevo Deutsch QC studies introduces the Qubits information in Universal Information qubits Quantum Turing Machine Reversibility 1994 Query 1981 complexity Shor invents Feynman calls for algorithm that Probability vs. a computer Quantum solves factoring mechanics based on QM for efficiently on a simulation Simon’s quantum algorithm purposes computer Factoring
The logic of quantum computing Timeline of QC
The Logic of 1970 2014 Quantum Computing
Marco Vitturi 1973 1985 Timeline of Holevo Deutsch QC studies introduces the Qubits information in Universal Information qubits Quantum Turing Machine Reversibility 1994 Query 1981 complexity Shor invents Feynman calls for algorithm that Probability vs. a computer Quantum solves factoring mechanics based on QM for efficiently on a simulation Simon’s quantum algorithm purposes computer: Factoring BOOM
The logic of quantum computing Timeline of QC
The Logic of 1992 2014 Quantum Computing
Marco Vitturi 1994 Timeline of Shor’s QC algorithm Qubits
Information
Reversibility
Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing Timeline of QC
The Logic of 1992 2014 Quantum Computing
Marco Vitturi 1994 Timeline of Shor’s QC algorithm Qubits First workshop on QC by InformationNIST Reversibility
Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing Timeline of QC
The Logic of 1992 2014 Quantum Computing
Marco Vitturi 1994 Timeline of Shor’s QC algorithm Qubits First workshop on QC by InformationNIST Reversibility
Query complexity 1995
Probability vs. Quantum Error Quantum Correction mechanics
Simon’s algorithm
Factoring
The logic of quantum computing Timeline of QC
The Logic of 1992 2014 Quantum Computing
Marco Vitturi 1994 Timeline of Shor’s QC algorithm Qubits First workshop on QC by InformationNIST Reversibility
Query complexity 1995
Probability vs. Quantum Error Quantum Correction mechanics First working quantum logic Simon’s algorithm gate
Factoring
The logic of quantum computing Timeline of QC
The Logic of 1992 2014 Quantum Computing
Marco Vitturi 1994 Timeline of Shor’s QC algorithm Qubits First workshop on QC by InformationNIST Reversibility
Query complexity 1995
Probability vs. Quantum Error Quantum Correction mechanics First working quantum logic Simon’s algorithm gate Factoring 1996 The logic of Grover’s algorithm: quantum computing search√ database in O( N) Timeline of QC
The Logic of 1992 2014 Quantum Computing
Marco Vitturi 1997 Timeline of First gates QC based on NMR Qubits
Information
Reversibility
Query complexity 1995
Probability vs. Quantum Error Quantum Correction mechanics First working quantum logic Simon’s algorithm gate Factoring 1996 The logic of Grover’s algorithm: quantum computing search√ database in O( N) Timeline of QC
The Logic of 1992 2014 Quantum Computing
Marco Vitturi 1997 Timeline of First gates QC based on NMR Qubits 1998 Information First experimental Reversibility demonstration (2 qubits) Query complexity 1995
Probability vs. Quantum Error Quantum Correction mechanics First working quantum logic Simon’s algorithm gate Factoring 1996 The logic of Grover’s algorithm: quantum computing search√ database in O( N) Timeline of QC
The Logic of 1992 2014 Quantum Computing
Marco Vitturi 1997 Timeline of First gates QC based on NMR Qubits 1998 Information First experimental Reversibility demonstration (2 qubits) Query complexity 2001 Probability vs. First run of Quantum Shor’s mechanics algorithm: Simon’s factors algorithm Factoring 1996 The logic of Grover’s algorithm: quantum computing search√ database in O( N) Timeline of QC
The Logic of 1992 2014 Quantum Computing
Marco Vitturi 1997 Timeline of First gates QC based on NMR Qubits 1998 Information First experimental Reversibility demonstration (2 qubits) Query complexity 2001 Probability vs. First run of Quantum Shor’s mechanics algorithm: Simon’s factors 15 algorithm Factoring 1996 The logic of Grover’s algorithm: quantum computing search√ database in O( N) Timeline of QC
The Logic of 1992 2014 Quantum Computing
Marco Vitturi 1997 Timeline of First gates QC based on NMR Qubits 1998 Information First experimental Reversibility demonstration (2 qubits) Query complexity 2001 Probability vs. First run of Quantum Shor’s mechanics algorithm: Simon’s factors algorithm 15 = 3 × 5 Factoring 1996 The logic of Grover’s algorithm: quantum computing search√ database in O( N) Timeline of QC
The Logic of 1992 2014 Quantum Computing
Marco Vitturi 1997 2003 Timeline of First gates Quantum QC based on NMR optical gates Qubits realized 1998 Information First experimental Reversibility demonstration (2 qubits) Query complexity 2001 Probability vs. First run of Quantum Shor’s mechanics algorithm: Simon’s factors algorithm 15 = 3 × 5 Factoring 1996 The logic of Grover’s algorithm: quantum computing search√ database in O( N) Timeline of QC
The Logic of 1992 2014 Quantum Computing
Marco Vitturi 1997 2003 Timeline of First gates Quantum QC based on NMR optical gates Qubits realized 2006 1998 Information 12 qubits First experimental Reversibility demonstration (2 qubits) Query complexity 2001 Probability vs. First run of Quantum Shor’s mechanics algorithm: Simon’s factors algorithm 15 = 3 × 5 Factoring 1996 The logic of Grover’s algorithm: quantum computing search√ database in O( N) Timeline of QC
The Logic of 1992 2014 Quantum Computing
Marco Vitturi 1997 2003 Timeline of First gates Quantum QC based on NMR optical gates Qubits realized 2006 1998 Information 12 qubits First experimental Reversibility demonstration (2 qubits) 2007 Query D-WAVE claims complexity 2001 28 qubits Probability vs. First run of Quantum Shor’s mechanics algorithm: Simon’s factors algorithm 15 = 3 × 5 Factoring 1996 The logic of Grover’s algorithm: quantum computing search√ database in O( N) Timeline of QC
The Logic of 1992 2014 Quantum Computing
Marco Vitturi 1997 2003 Timeline of First gates Quantum QC based on NMR optical gates Qubits realized 2006 1998 Information 12 qubits First experimental Reversibility demonstration (2 qubits) 2007 Query D-WAVE claims complexity 2001 28 qubits Probability vs. First run of 2008 Quantum Shor’s mechanics D-WAVE claims algorithm: 128 qubits Simon’s factors algorithm 15 = 3 × 5 Factoring 1996 The logic of Grover’s algorithm: quantum computing search√ database in O( N) Timeline of QC
The Logic of 1992 2014 Quantum Computing
Marco Vitturi 1997 2003 Timeline of First gates Quantum QC based on NMR optical gates Qubits realized 2006 1998 Information 12 qubits First experimental Reversibility demonstration (2 qubits) 2007 Query D-WAVE claims complexity 2001 28 qubits Probability vs. First run of 2008 Quantum Shor’s mechanics D-WAVE claims algorithm: 128 qubits Simon’s factors algorithm 2011 15 = 3 × 5 Factoring D-WAVE ONE 1996 made The logic of Grover’s algorithm: commercially quantum computing search√ database in available O( N) Timeline of QC
The Logic of 1992 2014 Quantum Computing
Marco Vitturi 1997 2003 2013 Timeline of First gates Quantum Quantum AI QC based on NMR optical gates Lab launched Qubits realized 2006 1998 Information 12 qubits First experimental Reversibility demonstration (2 qubits) 2007 Query D-WAVE claims complexity 2001 28 qubits Probability vs. First run of 2008 Quantum Shor’s mechanics D-WAVE claims algorithm: 128 qubits Simon’s factors algorithm 2011 15 = 3 × 5 Factoring D-WAVE ONE 1996 made The logic of Grover’s algorithm: commercially quantum computing search√ database in available O( N) Timeline of QC
The Logic of 1992 2014 Quantum Computing
Marco Vitturi 1997 2003 2013 Timeline of First gates Quantum Quantum AI QC based on NMR optical gates Lab launched Qubits realized 2006 1998 Information 12 qubits First experimental 2014 Reversibility demonstration (2 qubits) 2007 Snowden’s leaks Query D-WAVE claims reveal NSA very complexity 2001 28 qubits interested in Probability vs. First run of 2008 QC for cracking Quantum Shor’s mechanics D-WAVE claims purposes... algorithm: 128 qubits Simon’s factors algorithm 2011 15 = 3 × 5 Factoring D-WAVE ONE 1996 made The logic of Grover’s algorithm: commercially quantum computing search√ database in available O( N) If you measure it, you obtain 0 with probability |α|2, and 1 with probability |β|2.
Qubits
The Logic of Quantum Quantum unit of information: the qubit Computing A quantum observable with two possible values: Marco Vitturi
Timeline of α |0i + β |1i , with |α|2+|β|2= 1. QC
Qubits
Information
Reversibility
Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing Qubits
The Logic of Quantum Quantum unit of information: the qubit Computing A quantum observable with two possible values: Marco Vitturi
Timeline of α |0i + β |1i , with |α|2+|β|2= 1. QC
Qubits If you measure it, you obtain 0 with probability |α|2, and 1 with Information probability |β|2. Reversibility
Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing Qubits
The Logic of Quantum Quantum unit of information: the qubit Computing A quantum observable with two possible values: Marco Vitturi
Timeline of α |0i + β |1i , with |α|2+|β|2= 1. QC
Qubits If you measure it, you obtain 0 with probability |α|2, and 1 with Information probability |β|2. Reversibility
Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing Technical implementations
The Logic of Examples: Quantum Computing
Marco Vitturi
Timeline of QC
Qubits
Information
Reversibility
Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing Figure: Ions in an ion trap: |groundstatei , |excited statei Technical implementations
The Logic of Quantum Computing
Marco Vitturi
Timeline of QC
Qubits
Information
Reversibility
Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of Figure: polarization of a photon interacting with atoms trapped in a quantum cavity: |lefti , |righti computing Technical implementations
The Logic of Quantum Computing
Marco Vitturi
Timeline of QC
Qubits
Information
Reversibility
Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring The logic of Figure: nuclear spin of molecules measured with NMR: |↑i , |↓i quantum computing Technical implementations
The Logic of Quantum Computing
Marco Vitturi
Timeline of QC
Qubits
Information
Reversibility
Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring Figure: Superconducting QUantum Interference Device (SQUID) The logic of quantum computing 1 |1i • |1i |0i H √ (|0i + |1i) 2 |0i |1i Hadamard gate CNOT gate |1i R π i |1i × 2 × Phase shift gate SWAP gate An example of a circuit:
|ψi • • H H • •
|0i H U •
|0i H H •
Quantum circuits
The Logic of Quantum You can apply logic gates to qubits to perform computations Computing
Marco Vitturi
Timeline of QC
Qubits
Information
Reversibility
Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing |1i • |1i |0i |1i CNOT gate |1i R π i |1i × 2 × Phase shift gate SWAP gate An example of a circuit:
|ψi • • H H • •
|0i H U •
|0i H H •
Quantum circuits
The Logic of Quantum You can apply logic gates to qubits to perform computations Computing 1 Marco Vitturi √ (|0i + |1i) |0i H 2
Timeline of QC Hadamard gate
Qubits
Information
Reversibility
Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing |1i R π i |1i × 2 × Phase shift gate SWAP gate An example of a circuit:
|ψi • • H H • •
|0i H U •
|0i H H •
Quantum circuits
The Logic of Quantum You can apply logic gates to qubits to perform computations Computing 1 |1i • |1i Marco Vitturi |0i H √ (|0i + |1i) 2 |0i |1i Timeline of QC Hadamard gate CNOT gate Qubits
Information
Reversibility
Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing × × SWAP gate
An example of a circuit:
|ψi • • H H • •
|0i H U •
|0i H H •
Quantum circuits
The Logic of Quantum You can apply logic gates to qubits to perform computations Computing 1 |1i • |1i Marco Vitturi |0i H √ (|0i + |1i) 2 |0i |1i Timeline of QC Hadamard gate CNOT gate Qubits |1i R π i |1i Information 2
Reversibility Phase shift gate Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing An example of a circuit:
|ψi • • H H • •
|0i H U •
|0i H H •
Quantum circuits
The Logic of Quantum You can apply logic gates to qubits to perform computations Computing 1 |1i • |1i Marco Vitturi |0i H √ (|0i + |1i) 2 |0i |1i Timeline of QC Hadamard gate CNOT gate Qubits |1i R π i |1i × Information 2 × Reversibility Phase shift gate SWAP gate Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing Quantum circuits
The Logic of Quantum You can apply logic gates to qubits to perform computations Computing 1 |1i • |1i Marco Vitturi |0i H √ (|0i + |1i) 2 |0i |1i Timeline of QC Hadamard gate CNOT gate Qubits |1i R π i |1i × Information 2 × Reversibility Phase shift gate SWAP gate Query complexity
Probability vs. An example of a circuit: Quantum mechanics |ψi • • H H • • Simon’s algorithm
Factoring |0i H U •
The logic of quantum |0i H H • computing The Logic of Quantum Computing
Marco Vitturi
Timeline of QC
Qubits
Information
Reversibility
Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum Figure: A D-Wave computer. computing The Logic of Quantum Computing
Marco Vitturi
Timeline of QC
Qubits
Information
Reversibility
Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring Figure: The inside of a D-Wave computer. The quantum unit is at the bottom - it needs to be cooled down to 0.02 K◦ in order to work. The logic of quantum computing Lambda calculus Register machine Conway’s Game of Life Billiard ball computer, etc
Is quantum computing just another equivalent model to Turing machines?
THE question
The Logic of Quantum Computing Marco Vitturi There are many equivalent models of computation:
Timeline of QC Turing Machines
Qubits
Information
Reversibility
Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing Register machine Conway’s Game of Life Billiard ball computer, etc
Is quantum computing just another equivalent model to Turing machines?
THE question
The Logic of Quantum Computing Marco Vitturi There are many equivalent models of computation:
Timeline of QC Turing Machines Qubits Lambda calculus Information
Reversibility
Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing Conway’s Game of Life Billiard ball computer, etc
Is quantum computing just another equivalent model to Turing machines?
THE question
The Logic of Quantum Computing Marco Vitturi There are many equivalent models of computation:
Timeline of QC Turing Machines Qubits Lambda calculus Information Register machine Reversibility
Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing Billiard ball computer, etc
Is quantum computing just another equivalent model to Turing machines?
THE question
The Logic of Quantum Computing Marco Vitturi There are many equivalent models of computation:
Timeline of QC Turing Machines Qubits Lambda calculus Information Register machine Reversibility Conway’s Game of Life Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing Is quantum computing just another equivalent model to Turing machines?
THE question
The Logic of Quantum Computing Marco Vitturi There are many equivalent models of computation:
Timeline of QC Turing Machines Qubits Lambda calculus Information Register machine Reversibility Conway’s Game of Life Query complexity Billiard ball computer, etc Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing THE question
The Logic of Quantum Computing Marco Vitturi There are many equivalent models of computation:
Timeline of QC Turing Machines Qubits Lambda calculus Information Register machine Reversibility Conway’s Game of Life Query complexity Billiard ball computer, etc Probability vs. Quantum mechanics Is quantum computing just another equivalent model to Simon’s algorithm Turing machines? Factoring
The logic of quantum computing the status of n qubits is described by 2n complex numbers:
α0 |00 ... 00i + α1 |00 ... 01i + ... + α2n−1 |11 ... 1i
⇒ 32 qubits ≈ 4.3 billions complex numbers to be specified!
does it mean they contain much more information?
n qubits
The Logic of Quantum Computing Marco Vitturi you can compose qubits like bits: Timeline of QC |ψi ⊗ |φi = |ψ, φi Qubits
Information
Reversibility
Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing ⇒ 32 qubits ≈ 4.3 billions complex numbers to be specified!
does it mean they contain much more information?
n qubits
The Logic of Quantum Computing Marco Vitturi you can compose qubits like bits: Timeline of QC |ψi ⊗ |φi = |ψ, φi Qubits Information the status of n qubits is described by 2n complex numbers: Reversibility
Query n complexity α0 |00 ... 00i + α1 |00 ... 01i + ... + α2 −1 |11 ... 1i
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing does it mean they contain much more information?
n qubits
The Logic of Quantum Computing Marco Vitturi you can compose qubits like bits: Timeline of QC |ψi ⊗ |φi = |ψ, φi Qubits Information the status of n qubits is described by 2n complex numbers: Reversibility
Query n complexity α0 |00 ... 00i + α1 |00 ... 01i + ... + α2 −1 |11 ... 1i
Probability vs. Quantum ⇒ 32 qubits ≈ 4.3 billions complex numbers to be specified! mechanics
Simon’s algorithm
Factoring
The logic of quantum computing n qubits
The Logic of Quantum Computing Marco Vitturi you can compose qubits like bits: Timeline of QC |ψi ⊗ |φi = |ψ, φi Qubits Information the status of n qubits is described by 2n complex numbers: Reversibility
Query n complexity α0 |00 ... 00i + α1 |00 ... 01i + ... + α2 −1 |11 ... 1i
Probability vs. Quantum ⇒ 32 qubits ≈ 4.3 billions complex numbers to be specified! mechanics
Simon’s algorithm does it mean they contain much more information?
Factoring
The logic of quantum computing the reason is in Holevo bound [’73] You can extract at most n classical bits of information from n qubits.
⇒ qubits don’t convey more accessible information than classical bits do
Holevo’s bound
The Logic of Quantum Computing
Marco Vitturi
Timeline of QC The answer is NO.
Qubits
Information
Reversibility
Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing ⇒ qubits don’t convey more accessible information than classical bits do
Holevo’s bound
The Logic of Quantum Computing
Marco Vitturi
Timeline of QC The answer is NO.
Qubits
Information the reason is in Reversibility Holevo bound [’73] Query complexity You can extract at most n classical bits of information from n Probability vs. qubits. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing Holevo’s bound
The Logic of Quantum Computing
Marco Vitturi
Timeline of QC The answer is NO.
Qubits
Information the reason is in Reversibility Holevo bound [’73] Query complexity You can extract at most n classical bits of information from n Probability vs. qubits. Quantum mechanics
Simon’s ⇒ qubits don’t convey more accessible information than algorithm classical bits do Factoring
The logic of quantum computing this is Schr¨odingerequation in disguise, just differentiate: d −ih |ψi = H(t) |ψi dt Unitary ⇒ invertible ⇒ quantum computation is reversible!
typical logic gates aren’t: a AND b = 0 doesn’t allow to reconstruct a and b
Unitary evolution
The Logic of Quantum Computing
Marco Vitturi Evolution of an unobserved quantum system is unitary
Timeline of 0 0 ∗ QC |ψ(t )i = U(t , t) |ψ(t)i where UU = I Qubits
Information
Reversibility
Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing Unitary ⇒ invertible ⇒ quantum computation is reversible!
typical logic gates aren’t: a AND b = 0 doesn’t allow to reconstruct a and b
Unitary evolution
The Logic of Quantum Computing
Marco Vitturi Evolution of an unobserved quantum system is unitary
Timeline of 0 0 ∗ QC |ψ(t )i = U(t , t) |ψ(t)i where UU = I Qubits
Information this is Schr¨odingerequation in disguise, just differentiate: Reversibility d Query −ih |ψi = H(t) |ψi complexity dt Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing typical logic gates aren’t: a AND b = 0 doesn’t allow to reconstruct a and b
Unitary evolution
The Logic of Quantum Computing
Marco Vitturi Evolution of an unobserved quantum system is unitary
Timeline of 0 0 ∗ QC |ψ(t )i = U(t , t) |ψ(t)i where UU = I Qubits
Information this is Schr¨odingerequation in disguise, just differentiate: Reversibility d Query −ih |ψi = H(t) |ψi complexity dt Probability vs. Quantum Unitary ⇒ invertible ⇒ quantum computation is reversible! mechanics
Simon’s algorithm
Factoring
The logic of quantum computing Unitary evolution
The Logic of Quantum Computing
Marco Vitturi Evolution of an unobserved quantum system is unitary
Timeline of 0 0 ∗ QC |ψ(t )i = U(t , t) |ψ(t)i where UU = I Qubits
Information this is Schr¨odingerequation in disguise, just differentiate: Reversibility d Query −ih |ψi = H(t) |ψi complexity dt Probability vs. Quantum Unitary ⇒ invertible ⇒ quantum computation is reversible! mechanics
Simon’s algorithm typical logic gates aren’t: a AND b = 0 doesn’t allow to
Factoring reconstruct a and b
The logic of quantum computing input output 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 0 1 1 0 1 1 1 0 0 1 0 0 the Toffoli gate can re- 1 0 1 1 0 1 alize reversibly any other 1 1 0 1 1 1 gate 1 1 1 1 1 0
⇒ still no difference with classical computing
Toffoli gate
The Logic of Quantum Nevertheless, turns out you can implement classical Computing computation reversibly: Marco Vitturi
Timeline of QC
Qubits
Information
Reversibility
Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing ⇒ still no difference with classical computing
Toffoli gate
The Logic of Quantum Nevertheless, turns out you can implement classical Computing computation reversibly: Marco Vitturi
Timeline of input output QC 0 0 0 0 0 0 Qubits 0 0 1 0 0 1 Information 0 1 0 0 1 0 Reversibility 0 1 1 0 1 1 Query complexity 1 0 0 1 0 0 Probability vs. the Toffoli gate can re- 1 0 1 1 0 1 Quantum mechanics alize reversibly any other 1 1 0 1 1 1 Simon’s gate 1 1 1 1 1 0 algorithm
Factoring
The logic of quantum computing Toffoli gate
The Logic of Quantum Nevertheless, turns out you can implement classical Computing computation reversibly: Marco Vitturi
Timeline of input output QC 0 0 0 0 0 0 Qubits 0 0 1 0 0 1 Information 0 1 0 0 1 0 Reversibility 0 1 1 0 1 1 Query complexity 1 0 0 1 0 0 Probability vs. the Toffoli gate can re- 1 0 1 1 0 1 Quantum mechanics alize reversibly any other 1 1 0 1 1 1 Simon’s gate 1 1 1 1 1 0 algorithm
Factoring
The logic of quantum computing ⇒ still no difference with classical computing classical computing: N/2 queries on average √ quantum computing: only O( N) queries!
|0i H ···
|0i H ···
|0i H Q ∆ ··· Q ∆
|0i H ···
|0i H ···
Grover’s algorithm
The Logic of Quantum Computing Searching for an item in an unsorted, unstructured database: Marco Vitturi
Timeline of QC
Qubits
Information
Reversibility
Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing √ quantum computing: only O( N) queries!
|0i H ···
|0i H ···
|0i H Q ∆ ··· Q ∆
|0i H ···
|0i H ···
Grover’s algorithm
The Logic of Quantum Computing Searching for an item in an unsorted, unstructured database: Marco Vitturi classical computing: N/2 queries on average
Timeline of QC
Qubits
Information
Reversibility
Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing |0i H ···
|0i H ···
|0i H Q ∆ ··· Q ∆
|0i H ···
|0i H ···
Grover’s algorithm
The Logic of Quantum Computing Searching for an item in an unsorted, unstructured database: Marco Vitturi classical computing: N/2 queries on average √ Timeline of QC quantum computing: only O( N) queries!
Qubits
Information
Reversibility
Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing Grover’s algorithm
The Logic of Quantum Computing Searching for an item in an unsorted, unstructured database: Marco Vitturi classical computing: N/2 queries on average √ Timeline of QC quantum computing: only O( N) queries!
Qubits
Information
Reversibility |0i H ···
Query complexity |0i H ··· Probability vs. Quantum mechanics |0i H Q ∆ ··· Q ∆
Simon’s algorithm |0i H ··· Factoring
The logic of |0i H ··· quantum computing Example problems: determine MAJORITY of n inputs determine PARITY of n inputs evaluate trees of AND, OR collision problem (quantum algor. O(n1/3) queries) Sadly, no exponential speedup is possible (without some promise on the input): # classical queries = O (# quantum queries)6
Quantum queries
The Logic of Quantum Issue Computing The classical search requires only polynomially more queries. Marco Vitturi Can one do better than this? in particular, achieve a Timeline of QC logarithmic reduction in the number of queries?
Qubits
Information
Reversibility
Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing determine PARITY of n inputs evaluate trees of AND, OR collision problem (quantum algor. O(n1/3) queries) Sadly, no exponential speedup is possible (without some promise on the input): # classical queries = O (# quantum queries)6
Quantum queries
The Logic of Quantum Issue Computing The classical search requires only polynomially more queries. Marco Vitturi Can one do better than this? in particular, achieve a Timeline of QC logarithmic reduction in the number of queries?
Qubits Example problems: Information
Reversibility determine MAJORITY of n inputs
Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing evaluate trees of AND, OR collision problem (quantum algor. O(n1/3) queries) Sadly, no exponential speedup is possible (without some promise on the input): # classical queries = O (# quantum queries)6
Quantum queries
The Logic of Quantum Issue Computing The classical search requires only polynomially more queries. Marco Vitturi Can one do better than this? in particular, achieve a Timeline of QC logarithmic reduction in the number of queries?
Qubits Example problems: Information
Reversibility determine MAJORITY of n inputs Query determine PARITY of n inputs complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing collision problem (quantum algor. O(n1/3) queries) Sadly, no exponential speedup is possible (without some promise on the input): # classical queries = O (# quantum queries)6
Quantum queries
The Logic of Quantum Issue Computing The classical search requires only polynomially more queries. Marco Vitturi Can one do better than this? in particular, achieve a Timeline of QC logarithmic reduction in the number of queries?
Qubits Example problems: Information
Reversibility determine MAJORITY of n inputs Query determine PARITY of n inputs complexity
Probability vs. evaluate trees of AND, OR Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing Sadly, no exponential speedup is possible (without some promise on the input): # classical queries = O (# quantum queries)6
Quantum queries
The Logic of Quantum Issue Computing The classical search requires only polynomially more queries. Marco Vitturi Can one do better than this? in particular, achieve a Timeline of QC logarithmic reduction in the number of queries?
Qubits Example problems: Information
Reversibility determine MAJORITY of n inputs Query determine PARITY of n inputs complexity
Probability vs. evaluate trees of AND, OR Quantum 1/3 mechanics collision problem (quantum algor. O(n ) queries) Simon’s algorithm
Factoring
The logic of quantum computing Quantum queries
The Logic of Quantum Issue Computing The classical search requires only polynomially more queries. Marco Vitturi Can one do better than this? in particular, achieve a Timeline of QC logarithmic reduction in the number of queries?
Qubits Example problems: Information
Reversibility determine MAJORITY of n inputs Query determine PARITY of n inputs complexity
Probability vs. evaluate trees of AND, OR Quantum 1/3 mechanics collision problem (quantum algor. O(n ) queries) Simon’s algorithm Sadly, no exponential speedup is possible (without some
Factoring promise on the input):
The logic of quantum 6 computing # classical queries = O (# quantum queries) initialize the quantum register (qubits) apply a unitary operator composed of many smaller quantum gates measure the qubits at the end (collapsing to classical state)
Measurement is a probabilistic process ⇒ quantum algorithms are intrinsically probabilistic
so, are quantum computers just equivalent to Non-Deterministic Turing Machines?
Measurement is probabilistic
The Logic of Quantum Computing How quantum algorithms work: Marco Vitturi
Timeline of QC
Qubits
Information
Reversibility
Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing apply a unitary operator composed of many smaller quantum gates measure the qubits at the end (collapsing to classical state)
Measurement is a probabilistic process ⇒ quantum algorithms are intrinsically probabilistic
so, are quantum computers just equivalent to Non-Deterministic Turing Machines?
Measurement is probabilistic
The Logic of Quantum Computing How quantum algorithms work: Marco Vitturi initialize the quantum register (qubits) Timeline of QC
Qubits
Information
Reversibility
Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing measure the qubits at the end (collapsing to classical state)
Measurement is a probabilistic process ⇒ quantum algorithms are intrinsically probabilistic
so, are quantum computers just equivalent to Non-Deterministic Turing Machines?
Measurement is probabilistic
The Logic of Quantum Computing How quantum algorithms work: Marco Vitturi initialize the quantum register (qubits) Timeline of QC apply a unitary operator composed of many smaller Qubits quantum gates Information
Reversibility
Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing Measurement is a probabilistic process ⇒ quantum algorithms are intrinsically probabilistic
so, are quantum computers just equivalent to Non-Deterministic Turing Machines?
Measurement is probabilistic
The Logic of Quantum Computing How quantum algorithms work: Marco Vitturi initialize the quantum register (qubits) Timeline of QC apply a unitary operator composed of many smaller Qubits quantum gates Information
Reversibility measure the qubits at the end (collapsing to classical
Query state) complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing so, are quantum computers just equivalent to Non-Deterministic Turing Machines?
Measurement is probabilistic
The Logic of Quantum Computing How quantum algorithms work: Marco Vitturi initialize the quantum register (qubits) Timeline of QC apply a unitary operator composed of many smaller Qubits quantum gates Information
Reversibility measure the qubits at the end (collapsing to classical
Query state) complexity
Probability vs. Quantum Measurement is a probabilistic process ⇒ quantum mechanics algorithms are intrinsically probabilistic Simon’s algorithm
Factoring
The logic of quantum computing Measurement is probabilistic
The Logic of Quantum Computing How quantum algorithms work: Marco Vitturi initialize the quantum register (qubits) Timeline of QC apply a unitary operator composed of many smaller Qubits quantum gates Information
Reversibility measure the qubits at the end (collapsing to classical
Query state) complexity
Probability vs. Quantum Measurement is a probabilistic process ⇒ quantum mechanics algorithms are intrinsically probabilistic Simon’s algorithm so, are quantum computers just equivalent to Factoring
The logic of Non-Deterministic Turing Machines? quantum computing BQP: Bounded-error Quantum Polynomial time algorithms
P ⊆ BPP ⊆ BQP
reason: use a Hadamard gate to implement a coin flip!
√1 (|0i + |1i) |0i H 2
is it BPP ( BQP ? We don’t know.
BPP vs BQP
The Logic of Quantum First of all Computing
Marco Vitturi BPP: Bounded-error, Probabilistic, Polynomial time algorithms Timeline of QC
Qubits
Information
Reversibility
Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing P ⊆ BPP ⊆ BQP
reason: use a Hadamard gate to implement a coin flip!
√1 (|0i + |1i) |0i H 2
is it BPP ( BQP ? We don’t know.
BPP vs BQP
The Logic of Quantum First of all Computing
Marco Vitturi BPP: Bounded-error, Probabilistic, Polynomial time algorithms Timeline of QC BQP: Bounded-error Quantum Polynomial time Qubits algorithms Information
Reversibility
Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing reason: use a Hadamard gate to implement a coin flip!
√1 (|0i + |1i) |0i H 2
is it BPP ( BQP ? We don’t know.
BPP vs BQP
The Logic of Quantum First of all Computing
Marco Vitturi BPP: Bounded-error, Probabilistic, Polynomial time algorithms Timeline of QC BQP: Bounded-error Quantum Polynomial time Qubits algorithms Information Reversibility P ⊆ BPP ⊆ BQP Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing is it BPP ( BQP ? We don’t know.
BPP vs BQP
The Logic of Quantum First of all Computing
Marco Vitturi BPP: Bounded-error, Probabilistic, Polynomial time algorithms Timeline of QC BQP: Bounded-error Quantum Polynomial time Qubits algorithms Information Reversibility P ⊆ BPP ⊆ BQP Query complexity
Probability vs. reason: use a Hadamard gate to implement a coin flip! Quantum mechanics |0i √1 (|0i + |1i) Simon’s H 2 algorithm
Factoring
The logic of quantum computing We don’t know.
BPP vs BQP
The Logic of Quantum First of all Computing
Marco Vitturi BPP: Bounded-error, Probabilistic, Polynomial time algorithms Timeline of QC BQP: Bounded-error Quantum Polynomial time Qubits algorithms Information Reversibility P ⊆ BPP ⊆ BQP Query complexity
Probability vs. reason: use a Hadamard gate to implement a coin flip! Quantum mechanics |0i √1 (|0i + |1i) Simon’s H 2 algorithm
Factoring
The logic of quantum is it BPP BQP ? computing ( BPP vs BQP
The Logic of Quantum First of all Computing
Marco Vitturi BPP: Bounded-error, Probabilistic, Polynomial time algorithms Timeline of QC BQP: Bounded-error Quantum Polynomial time Qubits algorithms Information Reversibility P ⊆ BPP ⊆ BQP Query complexity
Probability vs. reason: use a Hadamard gate to implement a coin flip! Quantum mechanics |0i √1 (|0i + |1i) Simon’s H 2 algorithm
Factoring
The logic of quantum is it BPP BQP ? We don’t know. computing ( The Logic of Quantum Computing
Marco Vitturi
Timeline of QC
Qubits Information So is there any evidence of any advantage of quantum Reversibility computers over classical ones, or am I just wasting your time? Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing a classical algorithm requires Ω(2n/2) queries to f Simon’s quantum algorithm only O(n)! ⇒ Exponential speedup!
For some oracle A (a black-box function)
BPPA 6= BQPA
Simon’s algorithm
The Logic of Quantum Computing
Marco Vitturi Problem n n Timeline of You are given a boolean function f : {0, 1} → {0, 1} , QC “periodic”, i.e. ∃s, f(x) = f(x ⊕ s). Find s. Qubits
Information
Reversibility
Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing Simon’s quantum algorithm only O(n)! ⇒ Exponential speedup!
For some oracle A (a black-box function)
BPPA 6= BQPA
Simon’s algorithm
The Logic of Quantum Computing
Marco Vitturi Problem n n Timeline of You are given a boolean function f : {0, 1} → {0, 1} , QC “periodic”, i.e. ∃s, f(x) = f(x ⊕ s). Find s. Qubits
Information n/2 Reversibility a classical algorithm requires Ω(2 ) queries to f
Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing ⇒ Exponential speedup!
For some oracle A (a black-box function)
BPPA 6= BQPA
Simon’s algorithm
The Logic of Quantum Computing
Marco Vitturi Problem n n Timeline of You are given a boolean function f : {0, 1} → {0, 1} , QC “periodic”, i.e. ∃s, f(x) = f(x ⊕ s). Find s. Qubits
Information n/2 Reversibility a classical algorithm requires Ω(2 ) queries to f Query Simon’s quantum algorithm only O(n)! complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing For some oracle A (a black-box function)
BPPA 6= BQPA
Simon’s algorithm
The Logic of Quantum Computing
Marco Vitturi Problem n n Timeline of You are given a boolean function f : {0, 1} → {0, 1} , QC “periodic”, i.e. ∃s, f(x) = f(x ⊕ s). Find s. Qubits
Information n/2 Reversibility a classical algorithm requires Ω(2 ) queries to f Query Simon’s quantum algorithm only O(n)! ⇒ Exponential complexity speedup! Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing Simon’s algorithm
The Logic of Quantum Computing
Marco Vitturi Problem n n Timeline of You are given a boolean function f : {0, 1} → {0, 1} , QC “periodic”, i.e. ∃s, f(x) = f(x ⊕ s). Find s. Qubits
Information n/2 Reversibility a classical algorithm requires Ω(2 ) queries to f Query Simon’s quantum algorithm only O(n)! ⇒ Exponential complexity speedup! Probability vs. Quantum mechanics
Simon’s For some oracle A (a black-box function) algorithm Factoring BPPA 6= BQPA The logic of quantum computing Quantum programming at this level really resembles devising a physical experiment rather than actual programming.
Simon’s algorithm
The Logic of Quantum Computing
Marco Vitturi
Timeline of QC This is what the quantum circuit looks like
Qubits n ⊗n ⊗n Information |0 i H H
Reversibility Uf |0ni Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing Simon’s algorithm
The Logic of Quantum Computing
Marco Vitturi
Timeline of QC This is what the quantum circuit looks like
Qubits n ⊗n ⊗n Information |0 i H H
Reversibility Uf |0ni Query complexity Probability vs. Quantum programming at this level really resembles devising a Quantum mechanics physical experiment rather than actual programming. Simon’s algorithm
Factoring
The logic of quantum computing Shor’s algorithm There exists a quantum algorithm that factors a composite number N in O((log N)3) steps.
Best classical algorithm known (General Number Field Sieve) only achieves superpolynomial time:
1/3 2/3 O e1.9(log N) (log log N)
Factoring is believed to be in NP
Shor’s algorithm
The Logic of Quantum Computing Marco Vitturi Probably everybody knows about it:
Timeline of QC
Qubits
Information
Reversibility
Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing Best classical algorithm known (General Number Field Sieve) only achieves superpolynomial time:
1/3 2/3 O e1.9(log N) (log log N)
Factoring is believed to be in NP
Shor’s algorithm
The Logic of Quantum Computing Marco Vitturi Probably everybody knows about it:
Timeline of QC Shor’s algorithm Qubits There exists a quantum algorithm that factors a composite Information number N in O((log N)3) steps. Reversibility
Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing Factoring is believed to be in NP
Shor’s algorithm
The Logic of Quantum Computing Marco Vitturi Probably everybody knows about it:
Timeline of QC Shor’s algorithm Qubits There exists a quantum algorithm that factors a composite Information number N in O((log N)3) steps. Reversibility Query Best classical algorithm known (General Number Field Sieve) complexity only achieves superpolynomial time: Probability vs. Quantum mechanics 1/3 2/3 O e1.9(log N) (log log N) Simon’s algorithm
Factoring
The logic of quantum computing Shor’s algorithm
The Logic of Quantum Computing Marco Vitturi Probably everybody knows about it:
Timeline of QC Shor’s algorithm Qubits There exists a quantum algorithm that factors a composite Information number N in O((log N)3) steps. Reversibility Query Best classical algorithm known (General Number Field Sieve) complexity only achieves superpolynomial time: Probability vs. Quantum mechanics 1/3 2/3 O e1.9(log N) (log log N) Simon’s algorithm Factoring Factoring is believed to be in NP The logic of quantum computing A family picture
The Logic of Quantum Conjectured view of the situation: Computing
Marco Vitturi NP-hard Timeline of QC
Qubits
Information NP-complete Reversibility ? Query complexity NP Probability vs. Quantum mechanics
Simon’s algorithm BQP BPP P Factoring
The logic of quantum computing Classical computers use Boolean logic Consider statements A = diffraction pattern is observed B = electron passes through the upper slit C = electron passes through the lower slit Then experiment reveals
A AND (B OR C) = TRUE,
but (A AND B) OR (A AND C) = FALSE ! ⇒ distributivity law fails in quantum logic
Making sense
The Logic of Quantum Do classical computers and quantum computers rely on the Computing same logic? Marco Vitturi
Timeline of QC
Qubits
Information
Reversibility
Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing Consider statements A = diffraction pattern is observed B = electron passes through the upper slit C = electron passes through the lower slit Then experiment reveals
A AND (B OR C) = TRUE,
but (A AND B) OR (A AND C) = FALSE ! ⇒ distributivity law fails in quantum logic
Making sense
The Logic of Quantum Do classical computers and quantum computers rely on the Computing same logic? Classical computers use Boolean logic Marco Vitturi
Timeline of QC
Qubits
Information
Reversibility
Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing Then experiment reveals
A AND (B OR C) = TRUE,
but (A AND B) OR (A AND C) = FALSE ! ⇒ distributivity law fails in quantum logic
Making sense
The Logic of Quantum Do classical computers and quantum computers rely on the Computing same logic? Classical computers use Boolean logic Marco Vitturi Consider statements
Timeline of A = diffraction QC pattern is observed Qubits B = electron passes Information through the upper Reversibility slit Query complexity C = electron passes
Probability vs. through the lower Quantum slit mechanics
Simon’s algorithm
Factoring
The logic of quantum computing but (A AND B) OR (A AND C) = FALSE ! ⇒ distributivity law fails in quantum logic
Making sense
The Logic of Quantum Do classical computers and quantum computers rely on the Computing same logic? Classical computers use Boolean logic Marco Vitturi Consider statements
Timeline of A = diffraction QC pattern is observed Qubits B = electron passes Information through the upper Reversibility slit Query complexity C = electron passes
Probability vs. through the lower Quantum slit mechanics Then experiment reveals Simon’s algorithm A AND (B OR C) = TRUE, Factoring
The logic of quantum computing ⇒ distributivity law fails in quantum logic
Making sense
The Logic of Quantum Do classical computers and quantum computers rely on the Computing same logic? Classical computers use Boolean logic Marco Vitturi Consider statements
Timeline of A = diffraction QC pattern is observed Qubits B = electron passes Information through the upper Reversibility slit Query complexity C = electron passes
Probability vs. through the lower Quantum slit mechanics Then experiment reveals Simon’s algorithm A AND (B OR C) = TRUE, Factoring
The logic of quantum but (A AND B) OR (A AND C) = FALSE ! computing Making sense
The Logic of Quantum Do classical computers and quantum computers rely on the Computing same logic? Classical computers use Boolean logic Marco Vitturi Consider statements
Timeline of A = diffraction QC pattern is observed Qubits B = electron passes Information through the upper Reversibility slit Query complexity C = electron passes
Probability vs. through the lower Quantum slit mechanics Then experiment reveals Simon’s algorithm A AND (B OR C) = TRUE, Factoring
The logic of quantum but (A AND B) OR (A AND C) = FALSE ! computing ⇒ distributivity law fails in quantum logic The logic that models quantum mechanics correctly is not boolean logic but a lattice.
The lattice of (closed) subspaces of a Hilbert space H
V AND W becomes V ∩ W V OR W becomes V ⊕ W NOT V becomes V⊥.
Lattices
The Logic of Quantum Computing Birkhoff and Von Neumann first formalized the logic of Marco Vitturi quantum mechanics:
Timeline of QC
Qubits
Information
Reversibility
Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing The lattice of (closed) subspaces of a Hilbert space H
V AND W becomes V ∩ W V OR W becomes V ⊕ W NOT V becomes V⊥.
Lattices
The Logic of Quantum Computing Birkhoff and Von Neumann first formalized the logic of Marco Vitturi quantum mechanics:
Timeline of QC The logic that models quantum mechanics correctly is not Qubits boolean logic but a lattice. Information
Reversibility
Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing V AND W becomes V ∩ W V OR W becomes V ⊕ W NOT V becomes V⊥.
Lattices
The Logic of Quantum Computing Birkhoff and Von Neumann first formalized the logic of Marco Vitturi quantum mechanics:
Timeline of QC The logic that models quantum mechanics correctly is not Qubits boolean logic but a lattice. Information Reversibility The lattice of (closed) subspaces of a Hilbert space H Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing V OR W becomes V ⊕ W NOT V becomes V⊥.
Lattices
The Logic of Quantum Computing Birkhoff and Von Neumann first formalized the logic of Marco Vitturi quantum mechanics:
Timeline of QC The logic that models quantum mechanics correctly is not Qubits boolean logic but a lattice. Information Reversibility The lattice of (closed) subspaces of a Hilbert space H Query complexity
Probability vs. Quantum mechanics V AND W becomes V ∩ W
Simon’s algorithm
Factoring
The logic of quantum computing NOT V becomes V⊥.
Lattices
The Logic of Quantum Computing Birkhoff and Von Neumann first formalized the logic of Marco Vitturi quantum mechanics:
Timeline of QC The logic that models quantum mechanics correctly is not Qubits boolean logic but a lattice. Information Reversibility The lattice of (closed) subspaces of a Hilbert space H Query complexity
Probability vs. Quantum mechanics V AND W becomes V ∩ W Simon’s V OR W becomes V ⊕ W algorithm
Factoring
The logic of quantum computing Lattices
The Logic of Quantum Computing Birkhoff and Von Neumann first formalized the logic of Marco Vitturi quantum mechanics:
Timeline of QC The logic that models quantum mechanics correctly is not Qubits boolean logic but a lattice. Information Reversibility The lattice of (closed) subspaces of a Hilbert space H Query complexity
Probability vs. Quantum mechanics V AND W becomes V ∩ W Simon’s V OR W becomes V ⊕ W algorithm ⊥ Factoring NOT V becomes V .
The logic of quantum computing The Logic of Quantum Computing
Marco Vitturi
Timeline of QC
Qubits
Information Questions? Reversibility
Query complexity
Probability vs. Quantum mechanics
Simon’s algorithm
Factoring
The logic of quantum computing