Russell Childs, profile picture
Uploaded byRussell Childs
245 views

Dirac demo (quantum mechanics with C++). Please note: There is a problem with SlideShare updating documents, so there may be divergence between the code and the copied & pasted output.

The document demonstrates a C++ framework for quantum mechanics using matrix mechanics with various quantum states, operators, and gates. It includes examples of operations with qubits, inner and outer products, and quantum circuits, specifically focusing on complex exponential forms and tensor products. Additionally, the document outlines a measurement process performed on quantum states by different entities, Alice and Bob.

Embed presentation

Download to read offline
@copyright Russell John Childs, PhD, 11th March 2017 ======================================================================================== Simple demo of C++ framework allowing quantum mechanics to be done with Matrix Mechanics using Dirac Bras, Kets, inner, outer products, operators. ======================================================================================== Output of code (see later): =========================================== Usage model: =========================================== e^(i*theta): Polar notation cos theta + i sin theta a + i*b: Cartesian notation a + i b (e^(i*p/4))*Bra<3>{"010"}: Creates 3-qubit bra e^(i pi/4) <010| Ket<3>{"010"}: Creates 3-qubit ket |010> Bra<3>{"010"}*Ket<3>{"010"}: Inner product <010|010> = 1 Bra<3>{"010"}*Ket<3>{"110"}: Inner product <010|110> = 0 Bra<2>{"01"}*Ket<3>{"011"}: Inner product <01|011> = |1> Bra<2>{"01"}*Bra<3>{"011"}: tensor prod <01|<011| = <01011| Ket<2>{"01"}*Ket<3>{"011"}: tensor prod |01>|011> = |01011> cos(theta)*Ket<1>{"0"}*Bra<1>{"0"} - sin(theta)*Ket<1>{"0"}*Bra<1>{"1"}+ sin(theta)*Ket<1>{"1"}*Bra<1>{"0"}+ cos(theta)*Ket<1>{"1"}*Bra<1>{"1"}: Ry(2*theta) = -- -- | cos theta |0><0| -sin theta |0><1| | | sin theta |1><0| cos theta |1><1| | -- -- Passive rotation operator. Applied to alpha*Ket<1>{"0"} + beta*Ket<1>{"1"} = alpha |0> + beta |1> = -- -- | alpha | | beta | -- -- Ket<1>{"0"}*Bra<1>{"0"} & Ket<1>{"1"}*Bra<1>{"1"} - operator tensor product |0><0| & |1><1| = |01><01| (Ket<2>{"00"} >> Hadamard >> CNOT21) - the quantum circuit: |00> -- H -- CNOT21 = 1/sqrt_2(|00> + |11>), i.e. entangled (psi >> M >> bob) - performs a measurement on psi using the operator 'bob' =========================================== End of Usage model: =========================================== gates.X -> (1+i*0)|0><1| + (1+i*0)|1><0| gates.Y -> (0-i*1)|0><1| + (0+i*1)|1><0| gates.Z -> (1+i*0)|0><0| + (-1+i*0)|1><1| gates.H -> (4.32978e-17+i*0.707107)|0><0| + (4.32978e-17+i*0.707107)|0><1| + (-4.32978e-17+i*0.707107)|1><0| + (4.32978e- 17-i*0.707107)|1><1| gates.S -> (1+i*0)|0><0| + (0+i*1)|1><1| gates.T -> (1+i*0)|0><0| + (0.707107+i*0.707107)|1><1| CNOT = ((Ket<1>{"0"}*Bra<1>{"0"}&gates.I) + (Ket<1>{"1"}*Bra<1>{"1"}&gates.X)) -> (1+i*0)|00><00| + (1+i*0)|01><01| + (1+i*0)|10><11| + (1+i*0)|11><10| gates.controlled(gates.X) -> (1+i*0)|00><00| + (1+i*0)|01><01| + (1+i*0)|10><11| + (1+i*0)|11><10| ~((e^(i*pi/4))*Ket<1>{"0"}*Bra<1>{"1"}) = ~(e^ip/4 |0><1|) -> (0.707107-i*0.707107)|1><0| ~gates.S = ~((1+i*0)|0><0| + (0+i*1)|1><1|) -> (1+i*0)|0><0| + (0-i*1)|1><1| Psi<Dirac::K,1>({"0"}) = |0> -> (1+i*0)*|0> (Psi<Dirac::K,1>({"0"})>>gates.H) - single qubit quantum circuit |0> -- H -> (-4.32978e-17+i*0.707107)*|1> + (4.32978e- 17+i*0.707107)*|0> (Psi<Dirac::K,2>({"00"}) >> Hadamard) -> (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|11> + (-4.32978e- 17+i*0.707107)*|01> CNOT21 -> (1+i*0)|00><00| + (1+i*0)|01><11| + (1+i*0)|10><10| + (1+i*0)|11><01| (Psi<Dirac::K,2>({"00"}) >> Hadamard >> CNOT21) -> (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e- 17+i*0.707107)*|11> Bob -> (1+i*0)|00><00| + (1+i*0)|01><01| + (1+i*0)|10><10| + (1+i*0)|11><11| Alice -> (1+i*0)|00><00| + (1+i*0)|01><01| + (1+i*0)|10><10| + (1+i*0)|11><11| Measurements: psi = (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e-17+i*0.707107)*|11>, Bob : (6.12323e- 17+i*1)*|00>, Alice : (6.12323e-17+i*1)*|00>} psi = (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e-17+i*0.707107)*|11>, Bob : (-6.12323e- 17+i*1)*|11>, Alice : (-6.12323e-17+i*1)*|11>} psi = (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e-17+i*0.707107)*|11>, Bob : (6.12323e- 17+i*1)*|00>, Alice : (6.12323e-17+i*1)*|00>} psi = (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e-17+i*0.707107)*|11>, Bob : (-6.12323e- 17+i*1)*|11>, Alice : (-6.12323e-17+i*1)*|11>} psi = (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e-17+i*0.707107)*|11>, Bob : (-6.12323e- 17+i*1)*|11>, Alice : (-6.12323e-17+i*1)*|11>} psi = (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e-17+i*0.707107)*|11>, Bob : (6.12323e- 17+i*1)*|00>, Alice : (6.12323e-17+i*1)*|00>}
psi = (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e-17+i*0.707107)*|11>, Bob : (-6.12323e- 17+i*1)*|11>, Alice : (-6.12323e-17+i*1)*|11>} psi = (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e-17+i*0.707107)*|11>, Bob : (6.12323e- 17+i*1)*|00>, Alice : (6.12323e-17+i*1)*|00>} psi = (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e-17+i*0.707107)*|11>, Bob : (6.12323e- 17+i*1)*|00>, Alice : (6.12323e-17+i*1)*|00>} psi = (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e-17+i*0.707107)*|11>, Bob : (6.12323e- 17+i*1)*|00>, Alice : (6.12323e-17+i*1)*|00>} (10+20*i)*K{"1010"} -> (10+i*20)*|1010> (10+20*i)*B{"1010"} -> <1010|*(10+i*20) (e^(i*pi/4))*B{"1010"}*K{"1010"}*~(e^(i*pi/4)) -> 1+i*0 (K{"1010"}*~(e^(i*pi/4))*((e^(i*pi/4))*B{"1010"})) -> (1+i*0)|1010><1010| (K{"1010"}*~(e^(i*pi/4))*((e^(i*pi/4))*B{"1010"}))*(K{"1010"}*~(e^(i*pi/4))) -> (0.707107-i*0.707107)*|1010> (B{"1010"}*~(e^(i*pi/4)))*(K{"1010"}*~(e^(i*pi/4))*((e^(i*pi/4))*B{"1010"})) -> <1010|*(0.707107-i*0.707107) (K{"1010"}*~(e^(i*pi/4)))*(K{"1010"}*~(e^(i*pi/4))) -> (2.22045e-16-i*1)*|10101010> (B{"1010"}*~(e^(i*pi/4)))*(B{"1010"}*~(e^(i*pi/4))) -> <10101010|*(2.22045e-16-i*1) Bra<2>{"10"}*Ket<3>{"100"} -> (1+i*0)*|0> Bra<2>{"11"}*Ket<3>{"100"} -> (0+i*0)*|0> Bra<5>{"10011"}*Ket<3>{"100"} -> <11|*(1+i*0) (Bra<1>{"1"}*Bra<1>{"0"})*(Ket<1>{"1"}*Ket<1>{"0"}*Ket<1>{"0"}) -> (1+i*0)*|0> (Bra<1>{"1"}*Bra<1>{"1"})*(Ket<1>{"1"}*Ket<2>{"00"}) -> (0+i*0)*|0> Bra<1>{"1"}*(Ket<3>{"100"}*Bra<5>{"10011"})*(Bra<2>{"10"}*Bra<1>{"0"}) -> (1+i*0)|00><10011100| Ket<1>{"1"}*(Ket<3>{"100"}*Bra<5>{"10011"})*(Bra<2>{"10"}*Bra<1>{"0"}) -> (1+i*0)|1100><10011100| std::cos(pi/4)*Ket<1>{"0"} + std::sin(pi/4)*(e^(i*pi/4))*Ket<1>{"1"} ->(0.5+i*0.5)*|1> + (0.707107+i*0)*|0> ((elem*Ket<1>{"0"}*Bra<1>{"0"} - elem*Ket<1>{"0"}*Bra<1>{"1"} + (elem*Ket<1>{"1"}*Bra<1>{"0"} + elem*Ket<1>{"1"}*Bra<1>{"1"}))) -> (0.707107+i*0)|0><0| + (-0.707107+i*0)|0><1| + (0.707107+i*0)|1><0| + (0.707107+i*0)|1><1| 10*i -> +i*10, i*10 -> +i*10 (10*(-i))*(10*i) -> 100 abs(10*i) -> 10, abs(i*10) -> 10 10 +100*i -> 10+i*100, 2*(10 +100*i) -> 20+i*200, (10 +100*i)*3 -> 30+i*300, 4*C{10,100*i} -> 40+i*400, C{10,100*i}*5 -> 50+i*500 (2+3*i)*C{4,5*i}-> -7+i*22 ~(2+3*i)*C{2,3*i} -> 13+i*0 abs(~(2+3*i)*C{2,3*i}) -> 13 (e^(i*pi/4)).m_real -> 0.707107, abs((e^(i*pi/4)).m_real) -> 0.707107 (1000.0*e^(-i*pi/4)) -> 707.107-i*707.107, 1000*~(e^(i*pi/4)) -> 707.107-i*707.107 (e^(i*pi/4))*(e^(i*pi/4)) -> 2.22045e-16+i*1 ~(e^(i*pi/4))*(e^(i*pi/4)) -> 1+i*0 Code: int main(void) { using namespace QuantumGates; typedef QuantumGates::Ket<4> K; typedef QuantumGates::Bra<4> B; std::cout << "===========================================" << std::endl; std::cout << "Usage model:" << std::endl; std::cout << "===========================================" << std::endl; std::cout << "e^(i*theta): Polar notation cos theta + i sin theta" << std::endl; std::cout << "a + i*b: Cartesian notation a + i b" << std::endl; std::cout << "(e^(i*p/4))*Bra<3>{"010"}: Creates 3-qubit bra e^(i pi/4) " "<010|" << std::endl; std::cout << "Ket<3>{"010"}: Creates 3-qubit ket |010>" << std::endl; std::cout << "Bra<3>{"010"}*Ket<3>{"010"}: Inner product <010|010> = 1" << std::endl; std::cout << "Bra<3>{"010"}*Ket<3>{"110"}: Inner product <010|110> = 0" << std::endl; std::cout << "Bra<2>{"01"}*Ket<3>{"011"}: Inner product <01|011> = " "|1>" << std::endl; std::cout << "Bra<2>{"01"}*Bra<3>{"011"}: tensor prod <01|<011| " "= <01011|" << std::endl; std::cout << "Ket<2>{"01"}*Ket<3>{"011"}: tensor prod |01>|011> = " "|01011>" << std::endl; std::cout << "cos(theta)*Ket<1>{"0"}*Bra<1>{"0"} " "- sin(theta)*Ket<1>{"0"}*Bra<1>{"1"}" "+ sin(theta)*Ket<1>{"1"}*Bra<1>{"0"}" "+ cos(theta)*Ket<1>{"1"}*Bra<1>{"1"}: " << std::endl << " Ry(2*theta) = " << std::endl << " -- --" << std::endl<< " | cos theta |0><0| -sin theta |0><1| |"<< std::endl<< " | sin theta |1><0| cos theta |1><1| |"<< std::endl<< " -- --" << std::endl<<
" Passive rotation operator. Applied to" << std::endl << " alpha*Ket<1>{"0"} + beta*Ket<1>{"1"} " " = alpha |0> + beta |1> =" << std::endl<< " -- --" << std::endl<< " | alpha |" << std::endl<< " | beta |" << std::endl<< " -- --" << std::endl; std::cout << "Ket<1>{"0"}*Bra<1>{"0"} & Ket<1>{"1"}*Bra<1>{"1"} -" " operator tensor product |0><0| & |1><1| = |01><01|" << std::endl; std::cout << "(Ket<2>{"00"} >> Hadamard >> CNOT21) - the quantum circuit:" " |00> -- H -- CNOT21 = 1/sqrt_2(|00> + |11>), i.e. entangled" << std::endl; std::cout << "(psi >> M >> bob) - performs a measurement on psi " "using the operator 'bob' " << std::endl; std::cout << "===========================================" << std::endl; std::cout << "End of Usage model:" << std::endl; std::cout << "===========================================" << std::endl; auto gates = Gate::instance(); std::cout << "gates.X -> " << gates.X << std::endl; std::cout << "gates.Y -> " << gates.Y << std::endl; std::cout << "gates.Z -> " << gates.Z << std::endl; std::cout << "gates.H -> " << gates.H << std::endl; std::cout << "gates.S -> " << gates.S << std::endl; std::cout << "gates.T -> " << gates.T << std::endl; auto CNOT = ((Ket<1>{"0"}*Bra<1>{"0"}&gates.I) + (Ket<1>{"1"}*Bra<1>{"1"}&gates.X)); std::cout << "CNOT = ((Ket<1>{"0"}*Bra<1>{"0"}&gates.I) + " "(Ket<1>{"1"}*Bra<1>{"1"}&gates.X)) -> " << CNOT << std::endl; std::cout << "gates.controlled(gates.X) -> " << gates.controlled(gates.X) << std::endl; std::cout << "~((e^(i*pi/4))*Ket<1>{"0"}*Bra<1>{"1"}) " "= ~(e^ip/4 |0><1|) -> " << ~((e^(i*pi/4))*Ket<1>{"0"}*Bra<1>{1}) << std::endl; std::cout << "~gates.S = ~(" << gates.S << ") -> " << ~gates.S << std::endl; std::cout << "Psi<Dirac::K,1>({"0"}) = |0> -> " << Psi<Dirac::K,1>({"0"}) << std::endl; std::cout << "(Psi<Dirac::K,1>({"0"})>>gates.H) - single qubit quantum circuit" " |0> -- H -> " << (Psi<Dirac::K,1>({"0"})>>gates.H) << std::endl; auto Hadamard = ((Ket<1>{"0"}*Bra<1>{"0"}&gates.H) + (Ket<1>{"1"}*Bra<1>{"1"}&gates.I)); std::cout << "(Psi<Dirac::K,2>({"00"}) >> Hadamard) -> " << (Psi<Dirac::K,2>({"00"}) >> Hadamard) << std::endl; auto CNOT21 = Ket<2>{"00"}*Bra<2>{"00"} + Ket<2>{"01"}*Bra<2>{"11"} + Ket<2>{"10"}*Bra<2>{"10"} + Ket<2>{"11"}*Bra<2>{"01"}; std::cout << "CNOT21 -> "<< CNOT21 << std::endl; std::cout << "(Psi<Dirac::K,2>({"00"}) >> Hadamard >> CNOT21) -> " << (Psi<Dirac::K,2>({"00"}) >> Hadamard >> CNOT21) << std::endl; auto psi = (Psi<Dirac::K,2>({"00"}) >> Hadamard >> CNOT21); auto bob = ((Ket<1>{"0"}*Bra<1>{"0"} + Ket<1>{"1"}*Bra<1>{"1"}) & gates.I); auto alice = (gates.I &(Ket<1>{"0"}*Bra<1>{"0"} + Ket<1>{"1"}*Bra<1>{"1"})); std::cout << "Bob -> " << bob << std::endl; std::cout << "Alice -> " << alice << std::endl; std::cout << "Measurements:" << std::endl; for(unsigned i=0; i<10; ++i) { auto psi = (Psi<Dirac::K,2>({"00"}) >> Hadamard >> CNOT21); std::cout << "psi = " << psi << ", "; std::cout << "Bob : " << (psi >> M >> bob) << ", "; std::cout << "Alice : " << (psi >> M >> alice) << "} "; std::cout << std::endl; } std::cout << "(10+20*i)*K{"1010"} -> " << (10+20*i)*K{"1010"}<< std::endl; std::cout << "(10+20*i)*B{"1010"} -> " << (10+20*i)*B{"1010"}<< std::endl; std::cout << "(e^(i*pi/4))*B{"1010"}*K{"1010"}*~(e^(i*pi/4)) -> " << (e^(i*pi/4))*B{"1010"}*K{"1010"}*~(e^(i*pi/4)) << std::endl; std::cout << "(K{"1010"}*~(e^(i*pi/4))*((e^(i*pi/4))*B{"1010"})) -> " <<(K{"1010"}*~(e^(i*pi/4))*((e^(i*pi/4))*B{"1010"})) << std::endl; std::cout << "(K{"1010"}*~(e^(i*pi/4))*((e^(i*pi/4))*B{"1010"}))*"
"(K{"1010"}*~(e^(i*pi/4)))" " -> " << (K{"1010"}*~(e^(i*pi/4))*((e^(i*pi/4))*B{"1010"}))* (K{"1010"}*~(e^(i*pi/4))) << std::endl; std::cout << "(B{"1010"}*~(e^(i*pi/4)))*(K{"1010"}*~(e^(i*pi/4))*" "((e^(i*pi/4))*B{"1010"})) -> " << (B{"1010"}*~(e^(i*pi/4)))*(K{"1010"}*~(e^(i*pi/4))* ((e^(i*pi/4))*B{"1010"})) << std::endl; std::cout << "(K{"1010"}*~(e^(i*pi/4)))*(K{"1010"}*~(e^(i*pi/4))) -> " << (K{"1010"}*~(e^(i*pi/4)))*(K{"1010"}*~(e^(i*pi/4))) << std::endl; std::cout << "(B{"1010"}*~(e^(i*pi/4)))*(B{"1010"}*~(e^(i*pi/4))) -> " << (B{"1010"}*~(e^(i*pi/4)))*(B{"1010"}*~(e^(i*pi/4))) << std::endl; std::cout << "Bra<2>{"10"}*Ket<3>{"100"} -> " << Bra<2>{"10"}*Ket<3>{"100"} << std::endl; std::cout << "Bra<2>{"11"}*Ket<3>{"100"} -> " << Bra<2>{"11"}*Ket<3>{"100"} << std::endl; std::cout << "Bra<5>{"10011"}*Ket<3>{"100"} -> " << Bra<5>{"10011"}*Ket<3>{"100"} << std::endl; std::cout << "(Bra<1>{"1"}*Bra<1>{"0"})*(Ket<1>{"1"}*Ket<1>{"0"}*Ket<1>{"0"})" " -> " << (Bra<1>{"1"}*Bra<1>{"0"})*(Ket<1>{"1"}*Ket<1>{"0"}*Ket<1>{"0"}) << std::endl; std::cout << "(Bra<1>{"1"}*Bra<1>{"1"})*(Ket<1>{"1"}*Ket<2>{"00"}) " "-> " << (Bra<1>{"1"}*Bra<1>{"1"})*(Ket<1>{"1"}*Ket<2>{"00"}) << std::endl; std::cout << "Bra<1>{"1"}*(Ket<3>{"100"}*Bra<5>{"10011"})*" "(Bra<2>{"10"}*Bra<1>{"0"}) -> " << Bra<1>{"1"}*(Ket<3>{"100"}*Bra<5>{"10011"})*(Bra<2>{"10"}*Bra<1>{"0"}) << std::endl; std::cout << "Ket<1>{"1"}*(Ket<3>{"100"}*Bra<5>{"10011"})*" "(Bra<2>{"10"}*Bra<1>{"0"}) -> " << Ket<1>{"1"}*(Ket<3>{"100"}*Bra<5>{"10011"})* (Bra<2>{"10"}*Bra<1>{"0"}) << std::endl; std::cout << "std::cos(pi/4)*Ket<1>{"0"} + std::sin(pi/4)*(e^(i*pi/4))*" "Ket<1>{"1"} ->" << std::cos(pi/4)*Ket<1>{"0"} + std::sin(pi/4)*(e^(i*pi/4))*Ket<1>{"1"} << std::endl; auto elem = 1/std::sqrt(2); std::cout <<"((elem*Ket<1>{"0"}*Bra<1>{"0"} " "- elem*Ket<1>{"0"}*Bra<1>{"1"} " "+ (elem*Ket<1>{"1"}*Bra<1>{"0"} " "+ elem*Ket<1>{"1"}*Bra<1>{"1"})))" " -> " << ((elem*Ket<1>{"0"}*Bra<1>{"0"} - elem*Ket<1>{"0"}*Bra<1>{"1"}+ (elem*Ket<1>{"1"}*Bra<1>{"0"} + elem*Ket<1>{"1"}*Bra<1>{"1"}))) << std::endl; std::cout << "10*i -> " << 10*i << ", i*10 -> " << i*10 << std::endl; std::cout << "(10*(-i))*(10*i) -> " << (10*(-i))*(10*i) << std::endl; std::cout << "abs(10*i) -> " << abs(10*i) << ", abs(i*10) -> " << abs(i*10)<< std::endl; std::cout << "10 +100*i -> " << 10 +100*i << ", 2*(10 +100*i) -> "<< 2*(10 +100*i) << ", (10 +100*i)*3 -> " << (10 +100*i)*3 << ", 4*C{10,100*i} -> " << 4*C{10,100*i} << ", C{10,100*i}*5 -> " << C{10,100*i}*5 << std::endl; std::cout << "(2+3*i)*C{4,5*i}-> " << (2+3*i)*C{4,5*i} << std::endl; std::cout << "~(2+3*i)*C{2,3*i} -> " << ~(2+3*i)*C{2,3*i} << std::endl; std::cout << "abs(~(2+3*i)*C{2,3*i}) -> " << abs(~(2+3*i)*C{2,3*i}) << std::endl; auto x = e^(i*pi/4); std::cout << "(e^(i*pi/4)).m_real -> " << x.m_real << ", abs((e^(i*pi/4)).m_real) -> " << abs(x.m_imag)<<std::endl; std::cout << "(1000.0*e^(-i*pi/4)) -> " << (1000.0*e^(-i*pi/4)) << ", 1000*~(e^(i*pi/4)) -> " << 1000*~(e^(i*pi/4)) << std::endl; std::cout << "(e^(i*pi/4))*(e^(i*pi/4)) -> " << x*x << std::endl; std::cout << "~(e^(i*pi/4))*(e^(i*pi/4)) -> "<< ~x*x << std::endl;
return 0; }

More Related Content

PPT
Python tutorial
byAllsoftSolutions
 
PDF
What Have The Properties Ever Done For Us
byMiklós Martin
 
PDF
[D15] 最強にスケーラブルなカラムナーDBよ、Hadoopとのタッグでビッグデータの地平を目指せ！by Daisuke Hirama
byInsight Technology, Inc.
 
PDF
Parse Everything With Elixir
byGabriele Lana
 
PPTX
Ruby's Arrays and Hashes with examples
byNiranjan Sarade
 
PDF
Functional Gradient Boosting based on Residual Network Perception
byAtsushi Nitanda
 
PDF
Explain this!
byFabio Telles Rodriguez
 
PDF
The Ring programming language version 1.5.3 book - Part 69 of 184
byMahmoud Samir Fayed
 
Python tutorial
byAllsoftSolutions
 
What Have The Properties Ever Done For Us
byMiklós Martin
 
[D15] 最強にスケーラブルなカラムナーDBよ、Hadoopとのタッグでビッグデータの地平を目指せ！by Daisuke Hirama
byInsight Technology, Inc.
 
Parse Everything With Elixir
byGabriele Lana
 
Ruby's Arrays and Hashes with examples
byNiranjan Sarade
 
Functional Gradient Boosting based on Residual Network Perception
byAtsushi Nitanda
 
Explain this!
byFabio Telles Rodriguez
 
The Ring programming language version 1.5.3 book - Part 69 of 184
byMahmoud Samir Fayed
 

What's hot

PDF
2017 reviews for laowai career center china job scam
byLaowaiCareerCenter
 
PDF
Recentrer l'intelligence artificielle sur les connaissances
byMathieu d'Aquin
 
PPTX
Ruby Language: Array, Hash and Iterators
bySarah Allen
 
PDF
C4 4.5
byBGEsp1
 
PDF
The Ring programming language version 1.9 book - Part 69 of 210
byMahmoud Samir Fayed
 
DOC
Qhttnew
byNgô Diễm
 
PDF
Percona Live 4/15/15: Transparent sharding database virtualization engine (DVE)
byTesora
 
PDF
Intro to OTP in Elixir
byJesse Anderson
 
PDF
Oceans 2019 tutorial-geophysical-nav_7-updated
byFrancisco Curado-Teixeira
 
PDF
FINAL Composite
bySean Tingey
 
PDF
The ABCs of OTP
byJesse Anderson
 
PDF
機械学習モデルの判断根拠の説明
bySatoshi Hara
 
PDF
Poodr ch8-composition
byorga shih
 
PDF
POTENCIAS Y RADICALES
byEducación
 
PDF
Lenses and Prisms in Swift - Elviro Rocca - Codemotion Rome 2018
byCodemotion
 
2017 reviews for laowai career center china job scam
byLaowaiCareerCenter
 
Recentrer l'intelligence artificielle sur les connaissances
byMathieu d'Aquin
 
Ruby Language: Array, Hash and Iterators
bySarah Allen
 
C4 4.5
byBGEsp1
 
The Ring programming language version 1.9 book - Part 69 of 210
byMahmoud Samir Fayed
 
Qhttnew
byNgô Diễm
 
Percona Live 4/15/15: Transparent sharding database virtualization engine (DVE)
byTesora
 
Intro to OTP in Elixir
byJesse Anderson
 
Oceans 2019 tutorial-geophysical-nav_7-updated
byFrancisco Curado-Teixeira
 
FINAL Composite
bySean Tingey
 
The ABCs of OTP
byJesse Anderson
 
機械学習モデルの判断根拠の説明
bySatoshi Hara
 
Poodr ch8-composition
byorga shih
 
POTENCIAS Y RADICALES
byEducación
 
Lenses and Prisms in Swift - Elviro Rocca - Codemotion Rome 2018
byCodemotion
 

Viewers also liked

PDF
Simple shared mutex UML
byRussell Childs
 
PDF
Brochure Sistema de Información para administración de establecimientos comer...
bySotosoftco
 
PPTX
NoSQL - Hands on
byJohannes Hoppe
 
PDF
Interview uml design
byRussell Childs
 
PDF
Multithreaded sockets c++11
byRussell Childs
 
PDF
Recursion to iteration automation.
byRussell Childs
 
PDF
Design pattern to avoid downcasting
byRussell Childs
 
PDF
IBM Kinexa Prove It! C programming test results.
byRussell Childs
 
PDF
Full_resume_Dr_Russell_John_Childs
byRussell Childs
 
PDF
Shared_memory_hash_table
byRussell Childs
 
PPTX
3Com 7030-10136
bysavomir
 
PPTX
What have you learned from your audience feedback?
byLuke Randall
 
PDF
Content Strategy Report: LPCS
byldush
 
PPTX
雷諾數測試與計算
byShin-Ping Tian
 
PDF
Cuidados de enfermería ante el accidente ofídico
byMusete
 
PPTX
プロジェクト
by光稀 松本
 
PDF
Políticos [Maçonaria]
byDo outro lado da barricada
 
PPTX
Perfect places
byismael cabezudo
 
PDF
Poster1
byJulio Cesar Rodriguez Molano
 
PDF
Рекламный буклет налоговой системы "Свободная экономика"
byMichael Chernyshev
 
Simple shared mutex UML
byRussell Childs
 
Brochure Sistema de Información para administración de establecimientos comer...
bySotosoftco
 
NoSQL - Hands on
byJohannes Hoppe
 
Interview uml design
byRussell Childs
 
Multithreaded sockets c++11
byRussell Childs
 
Recursion to iteration automation.
byRussell Childs
 
Design pattern to avoid downcasting
byRussell Childs
 
IBM Kinexa Prove It! C programming test results.
byRussell Childs
 
Full_resume_Dr_Russell_John_Childs
byRussell Childs
 
Shared_memory_hash_table
byRussell Childs
 
3Com 7030-10136
bysavomir
 
What have you learned from your audience feedback?
byLuke Randall
 
Content Strategy Report: LPCS
byldush
 
雷諾數測試與計算
byShin-Ping Tian
 
Cuidados de enfermería ante el accidente ofídico
byMusete
 
プロジェクト
by光稀 松本
 
Políticos [Maçonaria]
byDo outro lado da barricada
 
Perfect places
byismael cabezudo
 
Poster1
byJulio Cesar Rodriguez Molano
 
Рекламный буклет налоговой системы "Свободная экономика"
byMichael Chernyshev
 

Similar to Dirac demo (quantum mechanics with C++). Please note: There is a problem with SlideShare updating documents, so there may be divergence between the code and the copied & pasted output.

PDF
MMath Paper, Canlin Zhang
bycanlin zhang
 
PDF
Mi ordenador, de mayor, quiere ser cuántico (curso acelerado de simulación ...
byFacultad de Informática UCM
 
PDF
Quantum Computation and the Stabilizer Formalism for Error Correction
byDaniel Bulhosa Solórzano
 
PDF
Metal-Insulator Transitions I.pdf
byTheAnalyzed
 
PDF
Quantum Cryptography
byMax Braun
 
PDF
Operators n dirac in qm
byAnda Tywabi
 
PDF
Smooth entropies a tutorial
bywtyru1989
 
PDF
Grovers Algorithm
byCaseyHaaland
 
PDF
Tensor train to solve stochastic PDEs
byAlexander Litvinenko
 
PDF
newfesformu.pdf eqrrqewewewqfewfqeqwgfrfvg4w5
byJesusMoral3
 
PDF
Mathematics of quantum mechanics bridge course calicut university
byAthulKrishna45
 
PDF
Ladder operator
byMehboobAlam59
 
PDF
M.Sc. Phy SII UIV Quantum Mechanics
byPankaj Nagpure, Shri Shivaji Science College, Amravati
 
PDF
Algebraic foundations of quantum computing
bysitric
 
PDF
Quantum simulation Mathematics and Python examples
byRaviSankar637310
 
PDF
Tensor Train data format for uncertainty quantification
byAlexander Litvinenko
 
PDF
Realizations, Differential Equations, Canonical Quantum Commutators And Infin...
byvcuesta
 
PPTX
Quantum computing intrtoduction workshop karim elgammal - at facebook devel...
byKarim Elgammal
 
PPT
lec14.ppt
byRai Saheb Bhanwar Singh College Nasrullaganj
 
PDF
quantum chemistry on quantum computer handson by Q# (2019/8/4@MDR Hongo, Tokyo)
byMaho Nakata
 
MMath Paper, Canlin Zhang
bycanlin zhang
 
Mi ordenador, de mayor, quiere ser cuántico (curso acelerado de simulación ...
byFacultad de Informática UCM
 
Quantum Computation and the Stabilizer Formalism for Error Correction
byDaniel Bulhosa Solórzano
 
Metal-Insulator Transitions I.pdf
byTheAnalyzed
 
Quantum Cryptography
byMax Braun
 
Operators n dirac in qm
byAnda Tywabi
 
Smooth entropies a tutorial
bywtyru1989
 
Grovers Algorithm
byCaseyHaaland
 
Tensor train to solve stochastic PDEs
byAlexander Litvinenko
 
newfesformu.pdf eqrrqewewewqfewfqeqwgfrfvg4w5
byJesusMoral3
 
Mathematics of quantum mechanics bridge course calicut university
byAthulKrishna45
 
Ladder operator
byMehboobAlam59
 
M.Sc. Phy SII UIV Quantum Mechanics
byPankaj Nagpure, Shri Shivaji Science College, Amravati
 
Algebraic foundations of quantum computing
bysitric
 
Quantum simulation Mathematics and Python examples
byRaviSankar637310
 
Tensor Train data format for uncertainty quantification
byAlexander Litvinenko
 
Realizations, Differential Equations, Canonical Quantum Commutators And Infin...
byvcuesta
 
Quantum computing intrtoduction workshop karim elgammal - at facebook devel...
byKarim Elgammal
 
lec14.ppt
byRai Saheb Bhanwar Singh College Nasrullaganj
 
quantum chemistry on quantum computer handson by Q# (2019/8/4@MDR Hongo, Tokyo)
byMaho Nakata
 

More from Russell Childs

PDF
spinor_quantum_simulator_user_guide_.pdf
byRussell Childs
 
PDF
String searching o_n
byRussell Childs
 
PDF
String searching o_n
byRussell Childs
 
PDF
String searching o_n
byRussell Childs
 
PDF
String searching
byRussell Childs
 
PDF
Permute
byRussell Childs
 
PDF
Permute
byRussell Childs
 
PDF
Feature extraction using adiabatic theorem
byRussell Childs
 
PDF
Feature extraction using adiabatic theorem
byRussell Childs
 
PDF
Wavelets_and_multiresolution_in_two_pages
byRussell Childs
 
PDF
Relativity 2
byRussell Childs
 
PDF
Full resume dr_russell_john_childs_2016
byRussell Childs
 
PDF
Interview C++11 code
byRussell Childs
 
PDF
Dynamic programming burglar_problem
byRussell Childs
 
PDF
K d tree_cpp
byRussell Childs
 
PDF
Algorithms devised for a google interview
byRussell Childs
 
PDF
IBM Kinexa Prove It! C++ programming test results.
byRussell Childs
 
PDF
2 mathematical challenge_analysis_design_and_results
byRussell Childs
 
PDF
1 mathematical challenge_problem
byRussell Childs
 
PDF
3 mathematical challenge_code
byRussell Childs
 
spinor_quantum_simulator_user_guide_.pdf
byRussell Childs
 
String searching o_n
byRussell Childs
 
String searching o_n
byRussell Childs
 
String searching o_n
byRussell Childs
 
String searching
byRussell Childs
 
Permute
byRussell Childs
 
Permute
byRussell Childs
 
Feature extraction using adiabatic theorem
byRussell Childs
 
Feature extraction using adiabatic theorem
byRussell Childs
 
Wavelets_and_multiresolution_in_two_pages
byRussell Childs
 
Relativity 2
byRussell Childs
 
Full resume dr_russell_john_childs_2016
byRussell Childs
 
Interview C++11 code
byRussell Childs
 
Dynamic programming burglar_problem
byRussell Childs
 
K d tree_cpp
byRussell Childs
 
Algorithms devised for a google interview
byRussell Childs
 
IBM Kinexa Prove It! C++ programming test results.
byRussell Childs
 
2 mathematical challenge_analysis_design_and_results
byRussell Childs
 
1 mathematical challenge_problem
byRussell Childs
 
3 mathematical challenge_code
byRussell Childs
 

Recently uploaded

PDF
Probiotic bacteria and their importance in aquaculture.pdf
byChinaSamy1
 
PDF
Virtualization 1: Virtual Machine, Hypervisor (Virtual Machine Monitor)
bymrethein98
 
PDF
Detection of an NH3 Absorption Band at 2.2μm on Europa
bySérgio Sacani
 
PDF
Age of exploration - ppt presentation_EAPT
byericaangelatoledo06
 
PPT
CLASSIFICATION-OF-POISONS.ppt ,
bypreetithakurr888
 
PPTX
Muscles_of_mastication_Anatomy of neck.
bybushra9akhtar786
 
PPTX
Cell Transport Powerpoint.pptx for high school students
byraobondalakunta25
 
PPTX
Cell cycle ( mitosis and meiosis)Msc.pptx
byPallabiBehera9
 
PPTX
Grade 9 - Science Fourth Quarter Physics
byModezaPonje
 
PPTX
forensic significance of blood and its examination.pptx
byAdrijaKrishna
 
PDF
Factors affecting Plant Health – Edaphic factors and Biotic Factors
byNistarini College, Purulia (W.B) India
 
PPT
Thermodynamics Chapter-2 by Cengel and Boles
byrasoolurfa
 
PPTX
Antigens, Haptens and epitopes in immunology.pptx
byGurvinder Kaur
 
PPTX
OECD 204 (Acute Dermal Toxicity) .pptx
byManjari36
 
PDF
Rapid diagnosis and contamination of seafood and aquaculture products
byGirija90
 
PDF
Bio - Molecules ( जैव अणु ) - PDF Notes - Irfanullah Mehar - JJ Sir Chemistry...
byWorld of Wisdom
 
PDF
Possible identification of the Luna 9 Moon landing site using a novel machine...
bySérgio Sacani
 
PPTX
Pharmaceutical Calculation Techniques and Terminologies (1).pptx
bymuhammedsajjadhussei
 
PPTX
History_Taking_and_Physical_Examination_of_Cardiovascular_System.pptx
byalirazzaghi137900
 
PDF
How common are oxygenic photosynthesis and large coal deposits on exoplanets?
bySérgio Sacani
 
Probiotic bacteria and their importance in aquaculture.pdf
byChinaSamy1
 
Virtualization 1: Virtual Machine, Hypervisor (Virtual Machine Monitor)
bymrethein98
 
Detection of an NH3 Absorption Band at 2.2μm on Europa
bySérgio Sacani
 
Age of exploration - ppt presentation_EAPT
byericaangelatoledo06
 
CLASSIFICATION-OF-POISONS.ppt ,
bypreetithakurr888
 
Muscles_of_mastication_Anatomy of neck.
bybushra9akhtar786
 
Cell Transport Powerpoint.pptx for high school students
byraobondalakunta25
 
Cell cycle ( mitosis and meiosis)Msc.pptx
byPallabiBehera9
 
Grade 9 - Science Fourth Quarter Physics
byModezaPonje
 
forensic significance of blood and its examination.pptx
byAdrijaKrishna
 
Factors affecting Plant Health – Edaphic factors and Biotic Factors
byNistarini College, Purulia (W.B) India
 
Thermodynamics Chapter-2 by Cengel and Boles
byrasoolurfa
 
Antigens, Haptens and epitopes in immunology.pptx
byGurvinder Kaur
 
OECD 204 (Acute Dermal Toxicity) .pptx
byManjari36
 
Rapid diagnosis and contamination of seafood and aquaculture products
byGirija90
 
Bio - Molecules ( जैव अणु ) - PDF Notes - Irfanullah Mehar - JJ Sir Chemistry...
byWorld of Wisdom
 
Possible identification of the Luna 9 Moon landing site using a novel machine...
bySérgio Sacani
 
Pharmaceutical Calculation Techniques and Terminologies (1).pptx
bymuhammedsajjadhussei
 
History_Taking_and_Physical_Examination_of_Cardiovascular_System.pptx
byalirazzaghi137900
 
How common are oxygenic photosynthesis and large coal deposits on exoplanets?
bySérgio Sacani
 

Dirac demo (quantum mechanics with C++). Please note: There is a problem with SlideShare updating documents, so there may be divergence between the code and the copied & pasted output.

  • 1.
    @copyright Russell JohnChilds, PhD, 11th March 2017 ======================================================================================== Simple demo of C++ framework allowing quantum mechanics to be done with Matrix Mechanics using Dirac Bras, Kets, inner, outer products, operators. ======================================================================================== Output of code (see later): =========================================== Usage model: =========================================== e^(i*theta): Polar notation cos theta + i sin theta a + i*b: Cartesian notation a + i b (e^(i*p/4))*Bra<3>{"010"}: Creates 3-qubit bra e^(i pi/4) <010| Ket<3>{"010"}: Creates 3-qubit ket |010> Bra<3>{"010"}*Ket<3>{"010"}: Inner product <010|010> = 1 Bra<3>{"010"}*Ket<3>{"110"}: Inner product <010|110> = 0 Bra<2>{"01"}*Ket<3>{"011"}: Inner product <01|011> = |1> Bra<2>{"01"}*Bra<3>{"011"}: tensor prod <01|<011| = <01011| Ket<2>{"01"}*Ket<3>{"011"}: tensor prod |01>|011> = |01011> cos(theta)*Ket<1>{"0"}*Bra<1>{"0"} - sin(theta)*Ket<1>{"0"}*Bra<1>{"1"}+ sin(theta)*Ket<1>{"1"}*Bra<1>{"0"}+ cos(theta)*Ket<1>{"1"}*Bra<1>{"1"}: Ry(2*theta) = -- -- | cos theta |0><0| -sin theta |0><1| | | sin theta |1><0| cos theta |1><1| | -- -- Passive rotation operator. Applied to alpha*Ket<1>{"0"} + beta*Ket<1>{"1"} = alpha |0> + beta |1> = -- -- | alpha | | beta | -- -- Ket<1>{"0"}*Bra<1>{"0"} & Ket<1>{"1"}*Bra<1>{"1"} - operator tensor product |0><0| & |1><1| = |01><01| (Ket<2>{"00"} >> Hadamard >> CNOT21) - the quantum circuit: |00> -- H -- CNOT21 = 1/sqrt_2(|00> + |11>), i.e. entangled (psi >> M >> bob) - performs a measurement on psi using the operator 'bob' =========================================== End of Usage model: =========================================== gates.X -> (1+i*0)|0><1| + (1+i*0)|1><0| gates.Y -> (0-i*1)|0><1| + (0+i*1)|1><0| gates.Z -> (1+i*0)|0><0| + (-1+i*0)|1><1| gates.H -> (4.32978e-17+i*0.707107)|0><0| + (4.32978e-17+i*0.707107)|0><1| + (-4.32978e-17+i*0.707107)|1><0| + (4.32978e- 17-i*0.707107)|1><1| gates.S -> (1+i*0)|0><0| + (0+i*1)|1><1| gates.T -> (1+i*0)|0><0| + (0.707107+i*0.707107)|1><1| CNOT = ((Ket<1>{"0"}*Bra<1>{"0"}&gates.I) + (Ket<1>{"1"}*Bra<1>{"1"}&gates.X)) -> (1+i*0)|00><00| + (1+i*0)|01><01| + (1+i*0)|10><11| + (1+i*0)|11><10| gates.controlled(gates.X) -> (1+i*0)|00><00| + (1+i*0)|01><01| + (1+i*0)|10><11| + (1+i*0)|11><10| ~((e^(i*pi/4))*Ket<1>{"0"}*Bra<1>{"1"}) = ~(e^ip/4 |0><1|) -> (0.707107-i*0.707107)|1><0| ~gates.S = ~((1+i*0)|0><0| + (0+i*1)|1><1|) -> (1+i*0)|0><0| + (0-i*1)|1><1| Psi<Dirac::K,1>({"0"}) = |0> -> (1+i*0)*|0> (Psi<Dirac::K,1>({"0"})>>gates.H) - single qubit quantum circuit |0> -- H -> (-4.32978e-17+i*0.707107)*|1> + (4.32978e- 17+i*0.707107)*|0> (Psi<Dirac::K,2>({"00"}) >> Hadamard) -> (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|11> + (-4.32978e- 17+i*0.707107)*|01> CNOT21 -> (1+i*0)|00><00| + (1+i*0)|01><11| + (1+i*0)|10><10| + (1+i*0)|11><01| (Psi<Dirac::K,2>({"00"}) >> Hadamard >> CNOT21) -> (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e- 17+i*0.707107)*|11> Bob -> (1+i*0)|00><00| + (1+i*0)|01><01| + (1+i*0)|10><10| + (1+i*0)|11><11| Alice -> (1+i*0)|00><00| + (1+i*0)|01><01| + (1+i*0)|10><10| + (1+i*0)|11><11| Measurements: psi = (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e-17+i*0.707107)*|11>, Bob : (6.12323e- 17+i*1)*|00>, Alice : (6.12323e-17+i*1)*|00>} psi = (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e-17+i*0.707107)*|11>, Bob : (-6.12323e- 17+i*1)*|11>, Alice : (-6.12323e-17+i*1)*|11>} psi = (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e-17+i*0.707107)*|11>, Bob : (6.12323e- 17+i*1)*|00>, Alice : (6.12323e-17+i*1)*|00>} psi = (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e-17+i*0.707107)*|11>, Bob : (-6.12323e- 17+i*1)*|11>, Alice : (-6.12323e-17+i*1)*|11>} psi = (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e-17+i*0.707107)*|11>, Bob : (-6.12323e- 17+i*1)*|11>, Alice : (-6.12323e-17+i*1)*|11>} psi = (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e-17+i*0.707107)*|11>, Bob : (6.12323e- 17+i*1)*|00>, Alice : (6.12323e-17+i*1)*|00>}
  • 2.
    psi = (4.32978e-17+i*0.707107)*|00>+ (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e-17+i*0.707107)*|11>, Bob : (-6.12323e- 17+i*1)*|11>, Alice : (-6.12323e-17+i*1)*|11>} psi = (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e-17+i*0.707107)*|11>, Bob : (6.12323e- 17+i*1)*|00>, Alice : (6.12323e-17+i*1)*|00>} psi = (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e-17+i*0.707107)*|11>, Bob : (6.12323e- 17+i*1)*|00>, Alice : (6.12323e-17+i*1)*|00>} psi = (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e-17+i*0.707107)*|11>, Bob : (6.12323e- 17+i*1)*|00>, Alice : (6.12323e-17+i*1)*|00>} (10+20*i)*K{"1010"} -> (10+i*20)*|1010> (10+20*i)*B{"1010"} -> <1010|*(10+i*20) (e^(i*pi/4))*B{"1010"}*K{"1010"}*~(e^(i*pi/4)) -> 1+i*0 (K{"1010"}*~(e^(i*pi/4))*((e^(i*pi/4))*B{"1010"})) -> (1+i*0)|1010><1010| (K{"1010"}*~(e^(i*pi/4))*((e^(i*pi/4))*B{"1010"}))*(K{"1010"}*~(e^(i*pi/4))) -> (0.707107-i*0.707107)*|1010> (B{"1010"}*~(e^(i*pi/4)))*(K{"1010"}*~(e^(i*pi/4))*((e^(i*pi/4))*B{"1010"})) -> <1010|*(0.707107-i*0.707107) (K{"1010"}*~(e^(i*pi/4)))*(K{"1010"}*~(e^(i*pi/4))) -> (2.22045e-16-i*1)*|10101010> (B{"1010"}*~(e^(i*pi/4)))*(B{"1010"}*~(e^(i*pi/4))) -> <10101010|*(2.22045e-16-i*1) Bra<2>{"10"}*Ket<3>{"100"} -> (1+i*0)*|0> Bra<2>{"11"}*Ket<3>{"100"} -> (0+i*0)*|0> Bra<5>{"10011"}*Ket<3>{"100"} -> <11|*(1+i*0) (Bra<1>{"1"}*Bra<1>{"0"})*(Ket<1>{"1"}*Ket<1>{"0"}*Ket<1>{"0"}) -> (1+i*0)*|0> (Bra<1>{"1"}*Bra<1>{"1"})*(Ket<1>{"1"}*Ket<2>{"00"}) -> (0+i*0)*|0> Bra<1>{"1"}*(Ket<3>{"100"}*Bra<5>{"10011"})*(Bra<2>{"10"}*Bra<1>{"0"}) -> (1+i*0)|00><10011100| Ket<1>{"1"}*(Ket<3>{"100"}*Bra<5>{"10011"})*(Bra<2>{"10"}*Bra<1>{"0"}) -> (1+i*0)|1100><10011100| std::cos(pi/4)*Ket<1>{"0"} + std::sin(pi/4)*(e^(i*pi/4))*Ket<1>{"1"} ->(0.5+i*0.5)*|1> + (0.707107+i*0)*|0> ((elem*Ket<1>{"0"}*Bra<1>{"0"} - elem*Ket<1>{"0"}*Bra<1>{"1"} + (elem*Ket<1>{"1"}*Bra<1>{"0"} + elem*Ket<1>{"1"}*Bra<1>{"1"}))) -> (0.707107+i*0)|0><0| + (-0.707107+i*0)|0><1| + (0.707107+i*0)|1><0| + (0.707107+i*0)|1><1| 10*i -> +i*10, i*10 -> +i*10 (10*(-i))*(10*i) -> 100 abs(10*i) -> 10, abs(i*10) -> 10 10 +100*i -> 10+i*100, 2*(10 +100*i) -> 20+i*200, (10 +100*i)*3 -> 30+i*300, 4*C{10,100*i} -> 40+i*400, C{10,100*i}*5 -> 50+i*500 (2+3*i)*C{4,5*i}-> -7+i*22 ~(2+3*i)*C{2,3*i} -> 13+i*0 abs(~(2+3*i)*C{2,3*i}) -> 13 (e^(i*pi/4)).m_real -> 0.707107, abs((e^(i*pi/4)).m_real) -> 0.707107 (1000.0*e^(-i*pi/4)) -> 707.107-i*707.107, 1000*~(e^(i*pi/4)) -> 707.107-i*707.107 (e^(i*pi/4))*(e^(i*pi/4)) -> 2.22045e-16+i*1 ~(e^(i*pi/4))*(e^(i*pi/4)) -> 1+i*0 Code: int main(void) { using namespace QuantumGates; typedef QuantumGates::Ket<4> K; typedef QuantumGates::Bra<4> B; std::cout << "===========================================" << std::endl; std::cout << "Usage model:" << std::endl; std::cout << "===========================================" << std::endl; std::cout << "e^(i*theta): Polar notation cos theta + i sin theta" << std::endl; std::cout << "a + i*b: Cartesian notation a + i b" << std::endl; std::cout << "(e^(i*p/4))*Bra<3>{"010"}: Creates 3-qubit bra e^(i pi/4) " "<010|" << std::endl; std::cout << "Ket<3>{"010"}: Creates 3-qubit ket |010>" << std::endl; std::cout << "Bra<3>{"010"}*Ket<3>{"010"}: Inner product <010|010> = 1" << std::endl; std::cout << "Bra<3>{"010"}*Ket<3>{"110"}: Inner product <010|110> = 0" << std::endl; std::cout << "Bra<2>{"01"}*Ket<3>{"011"}: Inner product <01|011> = " "|1>" << std::endl; std::cout << "Bra<2>{"01"}*Bra<3>{"011"}: tensor prod <01|<011| " "= <01011|" << std::endl; std::cout << "Ket<2>{"01"}*Ket<3>{"011"}: tensor prod |01>|011> = " "|01011>" << std::endl; std::cout << "cos(theta)*Ket<1>{"0"}*Bra<1>{"0"} " "- sin(theta)*Ket<1>{"0"}*Bra<1>{"1"}" "+ sin(theta)*Ket<1>{"1"}*Bra<1>{"0"}" "+ cos(theta)*Ket<1>{"1"}*Bra<1>{"1"}: " << std::endl << " Ry(2*theta) = " << std::endl << " -- --" << std::endl<< " | cos theta |0><0| -sin theta |0><1| |"<< std::endl<< " | sin theta |1><0| cos theta |1><1| |"<< std::endl<< " -- --" << std::endl<<
  • 3.
    " Passive rotationoperator. Applied to" << std::endl << " alpha*Ket<1>{"0"} + beta*Ket<1>{"1"} " " = alpha |0> + beta |1> =" << std::endl<< " -- --" << std::endl<< " | alpha |" << std::endl<< " | beta |" << std::endl<< " -- --" << std::endl; std::cout << "Ket<1>{"0"}*Bra<1>{"0"} & Ket<1>{"1"}*Bra<1>{"1"} -" " operator tensor product |0><0| & |1><1| = |01><01|" << std::endl; std::cout << "(Ket<2>{"00"} >> Hadamard >> CNOT21) - the quantum circuit:" " |00> -- H -- CNOT21 = 1/sqrt_2(|00> + |11>), i.e. entangled" << std::endl; std::cout << "(psi >> M >> bob) - performs a measurement on psi " "using the operator 'bob' " << std::endl; std::cout << "===========================================" << std::endl; std::cout << "End of Usage model:" << std::endl; std::cout << "===========================================" << std::endl; auto gates = Gate::instance(); std::cout << "gates.X -> " << gates.X << std::endl; std::cout << "gates.Y -> " << gates.Y << std::endl; std::cout << "gates.Z -> " << gates.Z << std::endl; std::cout << "gates.H -> " << gates.H << std::endl; std::cout << "gates.S -> " << gates.S << std::endl; std::cout << "gates.T -> " << gates.T << std::endl; auto CNOT = ((Ket<1>{"0"}*Bra<1>{"0"}&gates.I) + (Ket<1>{"1"}*Bra<1>{"1"}&gates.X)); std::cout << "CNOT = ((Ket<1>{"0"}*Bra<1>{"0"}&gates.I) + " "(Ket<1>{"1"}*Bra<1>{"1"}&gates.X)) -> " << CNOT << std::endl; std::cout << "gates.controlled(gates.X) -> " << gates.controlled(gates.X) << std::endl; std::cout << "~((e^(i*pi/4))*Ket<1>{"0"}*Bra<1>{"1"}) " "= ~(e^ip/4 |0><1|) -> " << ~((e^(i*pi/4))*Ket<1>{"0"}*Bra<1>{1}) << std::endl; std::cout << "~gates.S = ~(" << gates.S << ") -> " << ~gates.S << std::endl; std::cout << "Psi<Dirac::K,1>({"0"}) = |0> -> " << Psi<Dirac::K,1>({"0"}) << std::endl; std::cout << "(Psi<Dirac::K,1>({"0"})>>gates.H) - single qubit quantum circuit" " |0> -- H -> " << (Psi<Dirac::K,1>({"0"})>>gates.H) << std::endl; auto Hadamard = ((Ket<1>{"0"}*Bra<1>{"0"}&gates.H) + (Ket<1>{"1"}*Bra<1>{"1"}&gates.I)); std::cout << "(Psi<Dirac::K,2>({"00"}) >> Hadamard) -> " << (Psi<Dirac::K,2>({"00"}) >> Hadamard) << std::endl; auto CNOT21 = Ket<2>{"00"}*Bra<2>{"00"} + Ket<2>{"01"}*Bra<2>{"11"} + Ket<2>{"10"}*Bra<2>{"10"} + Ket<2>{"11"}*Bra<2>{"01"}; std::cout << "CNOT21 -> "<< CNOT21 << std::endl; std::cout << "(Psi<Dirac::K,2>({"00"}) >> Hadamard >> CNOT21) -> " << (Psi<Dirac::K,2>({"00"}) >> Hadamard >> CNOT21) << std::endl; auto psi = (Psi<Dirac::K,2>({"00"}) >> Hadamard >> CNOT21); auto bob = ((Ket<1>{"0"}*Bra<1>{"0"} + Ket<1>{"1"}*Bra<1>{"1"}) & gates.I); auto alice = (gates.I &(Ket<1>{"0"}*Bra<1>{"0"} + Ket<1>{"1"}*Bra<1>{"1"})); std::cout << "Bob -> " << bob << std::endl; std::cout << "Alice -> " << alice << std::endl; std::cout << "Measurements:" << std::endl; for(unsigned i=0; i<10; ++i) { auto psi = (Psi<Dirac::K,2>({"00"}) >> Hadamard >> CNOT21); std::cout << "psi = " << psi << ", "; std::cout << "Bob : " << (psi >> M >> bob) << ", "; std::cout << "Alice : " << (psi >> M >> alice) << "} "; std::cout << std::endl; } std::cout << "(10+20*i)*K{"1010"} -> " << (10+20*i)*K{"1010"}<< std::endl; std::cout << "(10+20*i)*B{"1010"} -> " << (10+20*i)*B{"1010"}<< std::endl; std::cout << "(e^(i*pi/4))*B{"1010"}*K{"1010"}*~(e^(i*pi/4)) -> " << (e^(i*pi/4))*B{"1010"}*K{"1010"}*~(e^(i*pi/4)) << std::endl; std::cout << "(K{"1010"}*~(e^(i*pi/4))*((e^(i*pi/4))*B{"1010"})) -> " <<(K{"1010"}*~(e^(i*pi/4))*((e^(i*pi/4))*B{"1010"})) << std::endl; std::cout << "(K{"1010"}*~(e^(i*pi/4))*((e^(i*pi/4))*B{"1010"}))*"
  • 4.
    "(K{"1010"}*~(e^(i*pi/4)))" " -> "<< (K{"1010"}*~(e^(i*pi/4))*((e^(i*pi/4))*B{"1010"}))* (K{"1010"}*~(e^(i*pi/4))) << std::endl; std::cout << "(B{"1010"}*~(e^(i*pi/4)))*(K{"1010"}*~(e^(i*pi/4))*" "((e^(i*pi/4))*B{"1010"})) -> " << (B{"1010"}*~(e^(i*pi/4)))*(K{"1010"}*~(e^(i*pi/4))* ((e^(i*pi/4))*B{"1010"})) << std::endl; std::cout << "(K{"1010"}*~(e^(i*pi/4)))*(K{"1010"}*~(e^(i*pi/4))) -> " << (K{"1010"}*~(e^(i*pi/4)))*(K{"1010"}*~(e^(i*pi/4))) << std::endl; std::cout << "(B{"1010"}*~(e^(i*pi/4)))*(B{"1010"}*~(e^(i*pi/4))) -> " << (B{"1010"}*~(e^(i*pi/4)))*(B{"1010"}*~(e^(i*pi/4))) << std::endl; std::cout << "Bra<2>{"10"}*Ket<3>{"100"} -> " << Bra<2>{"10"}*Ket<3>{"100"} << std::endl; std::cout << "Bra<2>{"11"}*Ket<3>{"100"} -> " << Bra<2>{"11"}*Ket<3>{"100"} << std::endl; std::cout << "Bra<5>{"10011"}*Ket<3>{"100"} -> " << Bra<5>{"10011"}*Ket<3>{"100"} << std::endl; std::cout << "(Bra<1>{"1"}*Bra<1>{"0"})*(Ket<1>{"1"}*Ket<1>{"0"}*Ket<1>{"0"})" " -> " << (Bra<1>{"1"}*Bra<1>{"0"})*(Ket<1>{"1"}*Ket<1>{"0"}*Ket<1>{"0"}) << std::endl; std::cout << "(Bra<1>{"1"}*Bra<1>{"1"})*(Ket<1>{"1"}*Ket<2>{"00"}) " "-> " << (Bra<1>{"1"}*Bra<1>{"1"})*(Ket<1>{"1"}*Ket<2>{"00"}) << std::endl; std::cout << "Bra<1>{"1"}*(Ket<3>{"100"}*Bra<5>{"10011"})*" "(Bra<2>{"10"}*Bra<1>{"0"}) -> " << Bra<1>{"1"}*(Ket<3>{"100"}*Bra<5>{"10011"})*(Bra<2>{"10"}*Bra<1>{"0"}) << std::endl; std::cout << "Ket<1>{"1"}*(Ket<3>{"100"}*Bra<5>{"10011"})*" "(Bra<2>{"10"}*Bra<1>{"0"}) -> " << Ket<1>{"1"}*(Ket<3>{"100"}*Bra<5>{"10011"})* (Bra<2>{"10"}*Bra<1>{"0"}) << std::endl; std::cout << "std::cos(pi/4)*Ket<1>{"0"} + std::sin(pi/4)*(e^(i*pi/4))*" "Ket<1>{"1"} ->" << std::cos(pi/4)*Ket<1>{"0"} + std::sin(pi/4)*(e^(i*pi/4))*Ket<1>{"1"} << std::endl; auto elem = 1/std::sqrt(2); std::cout <<"((elem*Ket<1>{"0"}*Bra<1>{"0"} " "- elem*Ket<1>{"0"}*Bra<1>{"1"} " "+ (elem*Ket<1>{"1"}*Bra<1>{"0"} " "+ elem*Ket<1>{"1"}*Bra<1>{"1"})))" " -> " << ((elem*Ket<1>{"0"}*Bra<1>{"0"} - elem*Ket<1>{"0"}*Bra<1>{"1"}+ (elem*Ket<1>{"1"}*Bra<1>{"0"} + elem*Ket<1>{"1"}*Bra<1>{"1"}))) << std::endl; std::cout << "10*i -> " << 10*i << ", i*10 -> " << i*10 << std::endl; std::cout << "(10*(-i))*(10*i) -> " << (10*(-i))*(10*i) << std::endl; std::cout << "abs(10*i) -> " << abs(10*i) << ", abs(i*10) -> " << abs(i*10)<< std::endl; std::cout << "10 +100*i -> " << 10 +100*i << ", 2*(10 +100*i) -> "<< 2*(10 +100*i) << ", (10 +100*i)*3 -> " << (10 +100*i)*3 << ", 4*C{10,100*i} -> " << 4*C{10,100*i} << ", C{10,100*i}*5 -> " << C{10,100*i}*5 << std::endl; std::cout << "(2+3*i)*C{4,5*i}-> " << (2+3*i)*C{4,5*i} << std::endl; std::cout << "~(2+3*i)*C{2,3*i} -> " << ~(2+3*i)*C{2,3*i} << std::endl; std::cout << "abs(~(2+3*i)*C{2,3*i}) -> " << abs(~(2+3*i)*C{2,3*i}) << std::endl; auto x = e^(i*pi/4); std::cout << "(e^(i*pi/4)).m_real -> " << x.m_real << ", abs((e^(i*pi/4)).m_real) -> " << abs(x.m_imag)<<std::endl; std::cout << "(1000.0*e^(-i*pi/4)) -> " << (1000.0*e^(-i*pi/4)) << ", 1000*~(e^(i*pi/4)) -> " << 1000*~(e^(i*pi/4)) << std::endl; std::cout << "(e^(i*pi/4))*(e^(i*pi/4)) -> " << x*x << std::endl; std::cout << "~(e^(i*pi/4))*(e^(i*pi/4)) -> "<< ~x*x << std::endl;
  • 5.