Quantum Computing 1 ✬ ✩ Quantum State Transformation Lect 3 Goutam Biswas ✫ ✪ Quantum Computing 2 ✬ ✩ Quantum State Transition Postulate It seems that nature does not allow arbitrary state transformation. Change of states (over time) of a closeda quantum mechanical system are caused by a specific class of transformations, mathematically known as unitary transformations. aBy a closed system we mean that no measurement is performed on the sys- tem. Lect 3 Goutam Biswas ✫ ✪ Quantum Computing 3 ✬ Unitary Transformation ✩ A unitary transformation U : H H is an • → isomorphism, where H an inner product space (Hilbert space). So U is a bijection that preserves inner • product. In our notation U : x y and if • | i 7→ | i x , x′ H, then | i | i ∈ < x , x′ >=< Ux , Ux′ >. | i | i | i | i Lect 3 Goutam Biswas ✫ ✪ Quantum Computing 4 ✬ ✩ Unitary Matrix A unitary transformation will be represented by a unitary matrix. We call a complex matrix U as unitary if its conjugate transpose (Hermitian transpose) U † (or U ∗) is its inverse i.e UU † = I. Lect 3 Goutam Biswas ✫ ✪ Quantum Computing 5 ✬ ✩ One Qubit Transformation 1 0 We encode 0 = 0 and 1 = 1 . • | i | i So an 1-qubit unitary transformation or • 1-qubit gate is a 2 2 unitary matrix U i.e. × U †U = I. 1 This means U † = U − and U † is also unitary. • Lect 3 Goutam Biswas ✫ ✪ Quantum Computing 6 ✬ ✩ Properties It is clear that any quantum transformation is reversible, as U x = y if and only if | i | i U † y = x . | i | i Lect 3 Goutam Biswas ✫ ✪ Quantum Computing 7 ✬ ✩ Properties A unitary operator U is linear by definition. • If a quantum state x is a superposition of • | i a1 x1 + + a x , then | i ··· n | ni U x = U(a1 x1 + + a x )= | i | i ··· n | ni a1U x1 + + a U x . | i ··· n | ni Lect 3 Goutam Biswas ✫ ✪ Quantum Computing 8 ✬ Properties ✩ The inner-product of U x and U y is • | i | i x U U y = x I y = x y . h | † | i h | | i h | i A unitary operator preserves the norm or • length of a state vector. It maps an orthonormal basis to another • orthonormal basis e.g. 1 √2 1 √2 H = / / : 0 , 1 + , . 1/√2 1/√2 − {| i | i} → {| i |−i} Lect 3 Goutam Biswas ✫ ✪ Quantum Computing 9 ✬ ✩ Properties We know that U †U = I = UU † i.e. u1j u2j 1 if i = j, [ u1i u2i uni ] = I = . · · · ij · · · · 0 if i = j. unj 6 So the columns of U are orthonormal . U † is also unitary, the columns of U † are also orthonormal. So the rows of U are so. Lect 3 Goutam Biswas ✫ ✪ Quantum Computing 10 ✬ Properties ✩ If U1 and U2 are two unitary operators on • the same space, their product U1U2 is also unitary: (U1U2)†U1U2 = U2†U1†U1U2 = U2†IU2 = I. If U1 and U2 are unitary operators on spaces • V1 and V2 respectively, then U1 U2 is an ⊗ unitary operator on V1 V2: ⊗ (U † U †)(U1 U2) = (U †U1) (U †U2)= 1 ⊗ 2 ⊗ 1 ⊗ 2 I1 I2. ⊗ Lect 3 Goutam Biswas ✫ ✪ Quantum Computing 11 ✬ ✩ Properties If U1 and U2 are unitary operators on spaces • V1 and V2 respectively, then U1 U2 is an ⊕ unitary operator on V1 V2. ⊕ U1 + U2 need not be unitary, even if U1 and • U2 are. kU need not be unitary, even if U is. • Lect 3 Goutam Biswas ✫ ✪ Quantum Computing 12 ✬ ✩ Cloning of Classical Bit We can use a CNOT gate to copy a classical bit. a a 0 0 a = a ⊕ CNOT Lect 3 Goutam Biswas ✫ ✪ Quantum Computing 13 ✬ ✩ No-Cloning Principle There is no unitary operator that can create clone of an arbitrary qubit state. This is essentially an outcome of linearity. Assume that there is a 2-qubit unitary • transformation U such that U a0 = aa for | i | i any qubit state a . | i Let a and b be orthogonal states. • | i | i Lect 3 Goutam Biswas ✫ ✪ Quantum Computing 14 ✬ No-Cloning Principle ✩ We have U a0 = aa and U b0 = bb . • | i | i | i | i Consider the state c0 , where • | i c = 1 ( a + b ). So c0 = 1 ( a0 + b0 ). | i √2 | i | i | i √2 | i | i 1 U c0 = (U a0 + U b0 ), by linearity | i √2 | i | i 1 = ( aa + bb ) by cloning. √2 | i | i Lect 3 Goutam Biswas ✫ ✪ Quantum Computing 15 ✬ ✩ No-Cloning Principle On the other hand, U c0 = cc , by cloning | i | i 1 1 = ( a + b ) ( a + b ), √2 | i | i ⊗ √2 | i | i 1 = ( aa + ab + ba + bb ). 2 | i | i | i | i It is a contradiction as. 1 ( aa + bb ) = 1( aa + ab + ba + bb ). √2 | i | i 6 2 | i | i | i | i Lect 3 Goutam Biswas ✫ ✪ Quantum Computing 16 ✬ ✩ Outer Product Given a ket vector x , its dual is a bra • | i vector x . h | The inner product of x and y is x y C. • | i | i h | i ∈ The outer product of x and y is x y . • | i | i | i h | Lect 3 Goutam Biswas ✫ ✪ Quantum Computing 17 ✬ ✩ Outer Product If we represent x as a column vector • | i (x1, ,x ), then x is the row vector ··· n h | [x1 x ], where x is the conjugate of x . ··· n i i The outer product x y , is the tensor • | i h | product x y . | i ⊗ h | Lect 3 Goutam Biswas ✫ ✪ Quantum Computing 18 ✬ ✩ Outer Products: 1-bit Base Vectors 1 1 1 0 1 0 0 0 = 1 0 = h i = | ih | 0 ⊗ h i 0 0 0 h 1 0 i 1 1 0 1 0 1 0 1 = 0 1 = h i = | ih | 0 ⊗ h i 0 0 0 h 0 1 i 0 0 1 0 0 0 1 0 = 1 0 = h i = | ih | 1 ⊗ h i 1 1 0 h 1 0 i 0 0 0 1 0 0 1 0 = 0 1 = h i = . | ih | 1 ⊗ h i 1 0 1 h 0 1 i Lect 3 Goutam Biswas ✫ ✪ Quantum Computing 19 ✬ ✩ 1-Bit Transformation Any 1-bit transformation is a linear combination of the outer products of vectors. a b = a( 0 0 ) + b( 0 1 ) + c( 1 0 ) + d( 1 1 ), c d | i h | | i h | | i h | | i h | Lect 3 Goutam Biswas ✫ ✪ Quantum Computing 20 ✬ ✩ Property of Outer Product Let x , y be two states of the state-space of a | i | i quantum system, then ( x x ) y = x x y , | i h | | i | i h | i = x y x . h | i| i The outer product x x projects a vector y to the subspace spanned| i h by| x . | i | i Lect 3 Goutam Biswas ✫ ✪ Quantum Computing 21 ✬ ✩ Example 1 1 ( 0 0 ) 0 + 1 | i h | √2 | i √2 | i 1 = (( 0 0 ) 0 + ( 0 0 ) 1 ) √2 | i h | | i | i h | | i 1 = ( 0 0 0 + 0 0 1 ) √2 | i h | i | i h | i 1 = 0 . √2 | i Note: 0 0 is not a unitary transformation. | i h | Lect 3 Goutam Biswas ✫ ✪ Quantum Computing 22 ✬ ✩ 1-qubit Unitary Transformations The first 1-qubit unitary transformation is the identity map, the 2 2 identity matrix that × keeps a qubit state unchanged. 1 0 . 0 1 Lect 3 Goutam Biswas ✫ ✪ Quantum Computing 23 ✬ 1-qubit Unitary Transformations ✩ Next 1-qubit unitary transformation is the • not gate. It should interchange the amplitudes of base vectors i.e. (a 0 + b 1 )= b 0 + a 1 . ¬ | i | i | i | i The Boolean transformation matrix for NOT • will do the job. 0 1 a b 0 1 = , where = 0 1 + 1 0 . | ih | | ih | 1 0 b a 1 0 Lect 3 Goutam Biswas ✫ ✪ Quantum Computing 24 ✬ ✩ 1-qubit Unitary Transformations 1-qubit not-gate is called X or σ or σ1. It is • x one of the three Pauli matrices. In Boolean logic, identity and not are the • only two possible 1-bit reversible gates. But the situation is different in case of • quantum gates. Lect 3 Goutam Biswas ✫ ✪ Quantum Computing 25 ✬ Unitary Transformations are Uncountable ✩ cos sin U = θ θ is a unitary operator. sin θ cos θ • − cos θ sin θ The adjoint of U is U † = − , • sin θ cos θ satisfying cos θ sin θ cos θ sin θ 1 0 UU † = − = . sin θ cos θ sin θ cos θ 0 1 • − So for every value of θ we have a unitary • gate. Lect 3 Goutam Biswas ✫ ✪ Quantum Computing 26 ✬ No Finite Set of Generators ✩ There are uncountably many unitary • transformations. So it is impossible to get a finite set of generators or universal transformation gates. However there are finite set of • transformations that can approximate any arbitrary transformation to any desired accuracy. Lect 3 Goutam Biswas ✫ ✪ Quantum Computing 27 ✬ Tensor Product of Transformations ✩ Multi-qubit transformations can be • expressed as linear combination of tensor products of 1-qubit or 2-qubit transformations. Let in an n-qubit system the transformations • U1, ,U are applied to qubits ··· k (1,i1), (i1 + 1,i2), , (ik 1,n) respectively. ··· − The combined transformation on n-qubits is U = U1 U . ⊗···⊗ k Lect 3 Goutam Biswas ✫ ✪ Quantum Computing 28 ✬ ✩ Qubit Gate Array U1 U3 U4 U6 I 1 O1 I2 O2 I 3 O3 I 4 O4 U2 U5 Lect 3 Goutam Biswas ✫ ✪ Quantum Computing 29 ✬ ✩ Tensor Product of Transformations The leftmost transformation is • U1 I I U2. ⊗ ⊗ ⊗ Second one is U3 I I. • ⊗ ⊗ Third one is U4 I U5. • ⊗ ⊗ Last transformation is U6 I I. • ⊗ ⊗ Lect 3 Goutam Biswas ✫ ✪ Quantum Computing 30 ✬ ✩ General Form of an 1-Qubit Operator Let U be an 1-qubit operator.