Lecture notes: Quantum gates in matrix and ladder operator forms

Transcription

1 Phys 7 Topics in Particles & Fields Spring 3 Lecture v.. Lecture notes: Quantum gates in matrix and ladder operator forms Jeffrey Yepez Department of Physics and Astronomy University of Hawai i at Manoa Watanabe Hall, 55 Correa Road Honolulu, Hawai i yepez@hawaii.edu yepez Dated: January 5, 3 Contents I. Singleton ladder operators II. Multiple objects A. Qubits B. Qubit number operators III. Fermionic ladder operators A. Jordan-Wigner transformation B. Matrix representation 3 IV. Representations of perpendicular quantum gates 4 A. Ladder operator representation 4. H H case 4. swap gate and entangling swap gate 6 3. H 3 H case 6 4. aswap gate and entangling aswap gate 7 References 8 I. SINGLETON LADDER OPERATORS There are two basic operators from which all other quantum operator are constructed. These operators are and, which by matrix multiplication generate and. A one in each slot what could be simpler? Each of these four operators carries physical significance. They are named for their function. Raising ladder operator: a σ iσ a Lowering ladder operator: a σ + iσ. b number particle operator: n number hole operator: h n a a. c a a +. d Operating on logical states qubit basis states, the singleton ladder operators give a Raise to a a Exclusion of s b a Exclusion of s c a Lower to, d where the state is called oblivion. Furthermore, operating on the logical states, the singleton number operators give n Exclusion of s 3a n Counts s 3b h Counts s 3c h Exclusion of s. 3d From the simple identity n + h 4 follows the anticommutation relation algebraically expressing the local exclusion principle a a + a a. 5 In this lecture we use. Finally, the observable number a bit of information is implicitly defined as the eigenvalue of n: n. 6 II. MULTIPLE OBJECTS A. Qubits Tensor product state the state of independent qubits: q i q q q Q i q q q Q used for a few qubits, Q 3 q q q Q numbered state, q i or, for all i,..., Q.

2 III FERMIONIC LADDER OPERATORS B. Qubit number operators Using the singleton number operator Q n, 7 we can generate the multiple qubit number operators. So, the two qubit number operators Q are expressed as the following tensor products of n with identity and n n 8a 8b 8c n n 9a 9b, 9c where denotes the identity matrix. Similarly, the three qubit number operators Q 3 are expressed as the following tensor products n 3 n a n 3 n b n 3 3 n. c For any system with Q qubits, the α th number operator, n α can be expressed in a way that depends on a single n placed at the α th position within the following tensor product: Q terms {}}{ n α n a }{{} α th term α n. b This identity represents the unfolding of the Q-qubit system number operator as a tensor product. III. FERMIONIC LADDER OPERATORS All quantum gate operations can be represented in terms of the fermionic qubit creation and qubit annihilation operators in the number representation, denoted a α and a α respectively. This approach serves as a general computational formulation applicable to any quantum algorithm. Acting on a system of Q qubits, a α and a α create and destroy a fermionic number variable at the αth qubit a α n... n α... n Q {, n α ɛ n n Q, n α { ɛ n n a α n... n α... n Q Q, n α, n α, where the phase factor is 3 ɛ P α i ni. 4 See page 7 of Ref. Fetter and Walecka, 97 for this way of determining ɛ used by condensed matter theorists. The fermionic ladder operators satisfy the anticommutation relations {a α, a β } δ αβ 5 {a α, a β } {a α, a β }. The number operator n α a αa α has eigenvalues of or in the number representation when acting on a pure state, corresponding to the αth qubit being in state or respectively. A. Jordan-Wigner transformation With the logical one state of a qubit, notice that σ z, so one can count the number of preceding bits that contribute to the overall phase shift due to fermionic bit exchange involving the ith qubit with tensor product operator, σz i ψ Ni ψ. The phase factor is determined by the number of bit crossings N i i k n k in the state ψ and where the Boolean number variables are n k [, ]. Hence, an annihilation operator is decomposed into a tensor product known as the Jordan-Wigner transformation Jordan and Wigner, 98 a i σ i z a Q i 6 for integer i [, Q]. That is, begin with the single annihilation operator a. Then, the ith fermionic annihilation operator is a system of Q qubits has a matrix representation that is expressible as the tensor product of i number of Pauli matrices, one single a, followed by Q i number of ones as follows: i Q a i a. 7 k k i+

3 A Jordan-Wigner transformation III FERMIONIC LADDER OPERATORS Since 7 is the tensor product of Q elements, each one a matrix, the resulting representation of a i is a matrix of size Q Q, as expected. Since all the components of a i are real i.e.,, or, the ith creation operator is simple enough to compute by just transposing 7, a i at. That 7 satisfies the usual anticommutation relations is straightforward to prove. First, using 7 and since σ3 and {a, a }, we know that i Q {a i, a i } {a, a } 8a k k i+ 8b k Q. 8c Similarly, {a i, a i } and {a i, a i } follow from the singleton anticommutators {a, a} and {a, a }, respectively. Second, and without loss of generality, for the case of i < j, we have {a i, a j }, i k σ 3 a i a k j k i+ j k i+ i {a, } k j k i+ a a k j+ k j+ a i + k σ 3 a i + a k j+ k j k i+ j k i+ a a k j+ k j+ 9a 9b 9c 9d since {a, }. Similarly, we know {a i, a j } and {a i, a j }. Thus, we arrive at the end of the proof by combining what we have learned from 8 and 9 B. Matrix representation In the basis where qubits q and q are ordered left to right q q, the creation operators are {a i, a j } δ ij {a i, a j } {a i, a j }, a a, a a. for any i and j. Since a and a have real components, the annihilation operators are the transposes of the matrices given in, 3

4 IV REPRESENTATIONS OF PERPENDICULAR QUANTUM GATES a a T and a a T : a a, a a. IV. REPRESENTATIONS OF PERPENDICULAR QUANTUM GATES A type of quantum logic gate useful for casting quantum algorithms in various computational physics applications is a conservative quantum gate. It is a -qubit universal quantum gate associated with perpendicular pairwise entanglement. A conservative quantum gate conserves the bit count in the number representation of the qubit system i.e. the total spin magnetization of a spin- system. If conservative quantum gates are used to model basic qubit-qubit interactions in a large qubit system, then the large scale dynamics of the qubit system is ultimately constrained by a number continuity equation, as was mentioned earlier. In the most general situation, it is sufficient to consider only a block diagonal matrix that has a sub-block, which causes entanglement and is a member of the special unitary group SU. We can neglect the overall phase factor because this does not affect the quantum dynamics and therefore our sub-block need not be a member of the more general unitary group U. If U is a member of SU, it can be parameterized using three real numbers, ξ, ζ, and ϑ, as follows U e iξ cos ϑ e iζ sin ϑ e iζ sin ϑ e iξ cos ϑ A B. C D We can represent a general conservative quantum logical gate by the 4 4 unitary matrix Υ A B C D. 3 E We choose this form for Υ because we want to entangle only two of the basis states, with, so as to conserve particle number, and that is why we call Υ a conservative quantum gate. The component in the topleft corner is set to unity because we do not want Υ to alter the vacuum state in any way. However, we may allow the component in the bottom-right corner to be arbitrary. We will see that the value of this component will depend on the particle statistics, reflecting whether quantum logic gates are used to model quantum gases with particles obeying Fermi statistics or not. A. Ladder operator representation It is instructive to work out the ladder operators in the Q case, where it is simple to write down the matrix representation. Remarkably, all the results carry over to the arbitrary size qubit systems with Q. Consider the following five quadratic operators: a a a a including the compound number operators n n n n. n n, 4 5 The conservative quantum gate 3 can be expressed in terms of the operators 4 and 5 given above: Υ + A n n + Ba a + Ca a + D n n + E n n 6a + A n + Ba a + Ca a + D n A + D E n n. 6b We would like to find the Hamiltonian, H say, associated with Υ. Letting z denote a complex parameter, we begin by parametrizing 6b in terms of z Υz e zh, 7 and then we solve for H. To do this, we series expand in the parameter z: Υz + zh + z H +. 8 There are two cases of interest: first when the Hamiltonian is idempotent, H H, then 8 reduces to Υz + e z H, 9 and second when H H but H 3 H and H 4 H, then 7 reduces to Υz + sinh z H + cosh z H. 3 These cases are worked out below. A remarkable feature of this approach to deriving is that the imposition of the idempotent or tri-idempotent constraint will gives us a novel way to derive the exchange properties associated with Fermi statistics.. H H case From 3 and 9, we can solve for H: H e z Υ e z A B C D E. 3 4

5 A Ladder operator representation IV REPRESENTATIONS OF PERPENDICULAR QUANTUM GATES Let us pick a new set of variables to simplify matters: A A e z C C e z δ E e z. B B e z D D e z 3a 3b 3c Then inserting 3 into 3, the Hamiltonian has the simple matrix and operator representation A B H C D, 33 δ and from this we deduce the operator form of the idempotent Hamiltonian H Ba a +Ca a +Dn n +A n n +δn n. 34 Next, inserting the new variables 3 into 3 and 6b, the matrix and operator representations for the conservative quantum logic gate become Υz e zh e z A + e z B e z B e z D + e z δ + [ + e z Ba a + Ca a 35a 35b + Dn n + A n n + δn n ]. 35c Since the Hamiltonian must be Hermitian, H H, we know that C B and δ δ, so δ must be a real valued number. Also, since the Hamiltonian is idempotent, H H, we get the additional constraint equations on the components: which admit the solutions: A A + B 35d A + D 35e D D + B, 35f A ± 4 B 35g D 4 B. 35h Then inserting 35g and 35h into 33 and 34, we can specify the idempotent Hamiltonian with only one free complex parameter: H ± p 4 B B B p 4 B δ 36a Ba a + B a a + 4 B n n + ± 4 B n n + δn n 36b Ba a + B a a B n ± 4 B n + δ n n. 36c The associated conservative quantum logic gate can also be rewritten by inserting 35g and 35h into 35a: Υz ez + ± ez p 4 B e z B e z B ez + ez p 4 B e z δ + [ + e z + Ba a + B a a 4 B n + 37a ± 4 B n + δ n n ]. 37b A useful special case occurs if we choose B e iξ : H e iξ eiξ δ 38a a a e iξ + a a e iξ n n + δ n n. 38b Since n a a and n a a, we can rewrite the idempotent Hamiltonian as follows: H a e iξ a a e iξ a + δ n n. 39 5

6 A Ladder operator representation IV REPRESENTATIONS OF PERPENDICULAR QUANTUM GATES Also, Υz ez + ez e iξ ez e iξ ez + e z δ + 4a [ + e z a e iξ a a e iξ a. swap gate and entangling swap gate +δ n n ]. 4b Finally, for z iπ we get the quantum swap gate Υiπ e iξ e iξ 4a δ a e iξ a a e iξ a δ n n. 4b For ξ and δ, 4a is a classical swap gate. To satisfy the unitary condition for our quantum logic gate, ΥΥ, we must restrict the real-valued component δ by the following constraint equation: δ, 4 which implies that either δ or δ. Then, our quantum swap gate 4a can be rewritten as: Υiπ e iξ e iξ, 43 ± where the plus sign applies for the δ case and the minus sign for the δ case. For z iπ we get the entangling swap gate Υ iπ + i i e iξ i e iξ + i 44a i δ + [ ] + i a e iξ a a e iξ a + δ n n. 44b 3. H 3 H case There exists an alternative Hamiltonian that is not idempotent but has a similar property at third order, H 3 H but H H and not an involution, i.e. H, which can generate a conservative quantum logic gate of the form 3. In this second case, the series expansion of the quantum gate 7 reduces to the form 3, which is Υz + cosh z H + sinh zh. where as in the previous case either δ or δ. This imposes the following constraint equations on the components: A B 46a A + D 46b D B, 46c Our approach will be to assume the Hamiltonian still has the form 33 and that its square has a diagonal matrix form: H A B B D δ A B B D δ 45a δ n n + n n + δn n 45b n + n + δ n n, 45c which admit the solutions: A ± B D B. 47a 47b 6

7 A Ladder operator representation IV REPRESENTATIONS OF PERPENDICULAR QUANTUM GATES Then, the Hamiltonian has the form H ± p B B B p B δ B a a + B a a B n n 48a ± B n n + δn n 48b and hence, using 3, the matrix representation of the conservative quantum gate becomes B a a + B a a B n ± B n + δn n, 48c Υz cosh z ± p B sinh z B sinh z B sinh z cosh z p B sinh z e z δ + + cosh z [n + n + δ n n ] + sinh z [B a a + B a a B n ± ] B n + δn n 49a 49b + sinh zb a a + sinh zb a a + cosh z B n + cosh z ± B n + [e z δ cosh z ] n n. 49c A useful special case occurs for B ie iξ. Then, H ie iξ ie iξ δ 5a ie iξ a a ie iξ a a + δn n 5b a + ie iξ a a ie iξ a n n + δn n. The quantum gate has the form: Υz cosh z ie iξ sinh z ie iξ sinh z cosh z e z δ + + i sinh z e iξ a a e iξ a a 5c 5a + cosh z n + n + [e z δ cosh z ] n n. 5b 4. aswap gate and entangling aswap gate Finally, for z iπ gate Υ iπ we get the asymmetric quantum e iξ e iξ 5 i δ + + e iξ a a e iξ a a n n + [i δ + ]n n. 53 For ξ and δ, 5 is the classical antisymmetric swap gate. For z iπ 4 we get the entangling aswap gate 7

8 A Ladder operator representation IV REPRESENTATIONS OF PERPENDICULAR QUANTUM GATES Υ iπ 4 e iξ e iξ e iπ 4 δ + + e iξ a a e iξ a a + n + n n n + e iπ4 δn n. 54a 54b References Fetter, A. L., and J. D. Walecka, 97, Quantum Theory of Many-Particle Systems, International series in pure and applied physics McGraw-Hill Book Company, New York. Jordan, P., and E. Wigner, 98, Zeitschrift fur Physik A 479-, 63. 8