10 Quantum Complexity Theory I Just as the theory of computability had its foundations in the Church-Turing thesis, computa- tional complexity theory rests upon a modern strengthening of this thesis, which asserts that any \reasonable" model of computation can be e±ciently simulated on a probabilistic Turing machine (by \e±cient" we mean here a runtime that is bounded by a polynomial in the runtime of the simulated machine). For example, computers that can operate on arbitrary length words in unit time, or that can exactly compute real numbers with in¯nite precision are unreasonable models, since it seems clear that they cannot be physically implemented. It had been argued that the Turing machine model is the inevitable choice once we assume that we can implement only ¯nite precision computational primitives. Given the widespread belief that NP 6= BPP, this would seem to put a wide range of important computational problems (the NP-hard problems) well beyond the capability of computers. However, the Turing machine is an inadequate model for all physically realizable computing devices for a fundamental reason: the Turing machine is based on a classical physics model of the universe, whereas current physical theory asserts that the universe is quantum physical. Can we get inherently new kinds of (discrete) computing devices based on quantum physics? The ¯rst indication that such a device might potentially be more powerful than a probabilistic Turing ma- chine appeared in a paper by Feynman about two decades ago. In that paper, Feynman pointed out a very curious problem: it appears to be impossible to simulate a general quantum physical system on a probabilistic Turing machine without an exponential slowdown. The di±culty with the simulation has nothing to do with the problem of simulating a continuous system with a discrete one { we may assume that the quantum physical system to be simulated is discrete, some kind of quantum cellular automaton. It has rather to do with a phenomenon in quan- tum physics that allows di®erent outcomes of a quantum physical e®ect to interfere with each other in a strange way. In view of Feynman's observation, we must re-examine the foundations of computational complexity theory and the complexity-theoretic form of the Church-Turing thesis, and study the computational power of computing devices based on quantum physics. A ¯rst precise model of a quantum physical computer, called the quantum Turing machine, was formulated by Deutsch. This model may be thought of as a quantum physical analogue of a probabilistic Turing machine: it has an in¯nite tape and a ¯nite state control, and the actions of the machine are local and completely speci¯ed by this ¯nite state control. Furthermore, on a given input a quantum Turing machine produces a random outcome according to a probability distribution. However, instead of a probabilistic Turing machine, the quantum Turing machine can generate probability distributions in which the amplitudes of certain con¯gurations can have complex instead of just positive real numbers. 10.1 Mathematical foundations In order to understand how quantum computers work, we need some background in linear algebra and complex numbers. Complex numbers The set of complex numbers IC is de¯ned as the set IR £ IR with the following two operations: 1 ² For all (a; b); (a0; b0) 2 IC,(a; b) + (a0; b0) = (a + a0; b + b0). ² For all (a; b); (a0; b0) 2 IC,(a; b) ¢ (a0; b0) = (a ¢ a0 ¡ b ¢ b0; a ¢ b0 + b ¢ a0). p When representing (1; 0) as 1 and (0; 1) as the imaginary number i = ¡1, we can also write every number z 2 IC as z = a + ib for some a; b 2 IR, and we can use the standard addition and multiplication operations on these numbers. This is correct because for any z = a + ib and z0 = a0 + ib0 it follows that ² z + z0 = (a + a0) + i(b + b0) and ² z ¢ z0 = (a + ib)(a0 + ib0) = aa0 + iab0 + iba0 + i2bb0 = (aa0 ¡ bb0) + i(ab0 + ba0), which matches the rules for addition and multiplication above. There is yet another way of representingp a complex number. Any z = a+ib can be viewed as a 2-dimensional vector of length ` = a2 + b2 and angle ® = tan(b=a). Since the Euler function extended to the complex numbers has the property that ei® = cos ® + i sin ® it follows that z can also be represented as ` ¢ ei®. This immediately implies that z2 = (` ¢ ei®)2 = `2 ¢ ei2® 2 2 pi.e., z is a 2-dimensional vector of lengthp` and angle 2®. On the other hand, this implies that z isp a 2-dimensional vector of length ` and angle ®=p2. Hence, if we set z = (¡1; 0) = ¡1, then z = (0; 1) = i, and so it is natural to de¯ne i = ¡1 above. Given a complex number z = a + ib, ² z¤ = a ¡ ib is called the complex conjugate of z and ² jjzjj2 = z ¢ z¤ = a2 + b2 is called the square norm of z Linear algebra We start with linear algebra over the real numbers. Let M(m; n; IR) be the space of all m £ n- dimensional matrices 0 1 a11 a12 ¢ ¢ ¢ a1n B C B a a ¢ ¢ ¢ a C B 21 22 2n C A = B . C @ . A am1 am2 ¢ ¢ ¢ amn with aij 2 IR for every 1 · i · m and 1 · j · n. If n = 1, we just call A a vector and denote it by a instead of using a capital letter. When m and n are clear from the context, then we also write A = (aij) and a = (ai). Addition and multiplication over matrices is de¯ned as follows: ² For all A; B 2 M(m; n; IR), A + B = C 2 M(m; n; IR) with cij = aij + bij for all i; j. ² For all A 2 M(m; n; IR) and B 2 M(n; p; IR), A ¢ B = C 2 M(m; p; IR) with cij = Pm k=1 aik ¢ bkj for all i; j. 2 T 0 Given a matrix A 2 M(m; n; IR), the transpose A = (aij) 2 M(n; m; IR) of A is de¯ned as 0 aij = aji for all i; j. n Given a vector a 2 IR , its `1-norm is de¯ned as jaj = ja1j + ja2j + ::: + janj ; its `2-norm or Euclidean length is de¯ned as q 2 2 2 jjajj = a1 + a2 + ::: + an and its square norm is de¯ned as jjajj2. When applying the matrix product to vectors, we obtain some interesting insights. Given two vectors a; b 2 IRn, it holds for the angle ® between a and b that aT ¢ b cos ® = jjajj ¢ jjbjj Hence, (aT ¢ b)=jjajj is equal to the length of the projection of b onto a. A projection can be seen as the shadow of one vector on another. As an example, if the two vectors are orthogonal to each other, i.e., they form a right angle, then their scalar product is 0, but if the two vectors are parallel to each other, then (aT ¢ b)=jjajj = jjbjj. Orthogonal vectors are also called linearly independent. The identity matrix In = (eij) 2 M(n; n; IR) is the matrix with 1's along the diagonal and all other entries being 0, i.e., eii = 1 for all i and eij = 0 for all i 6= j. The vector ek = (ei) has entries that are de¯ned as ek = 1 and ei = 0 for all i 6= k. A subset U ⊆ M(m; n; IR) is called a subspace of M(m; n; IR) if ² U 6= ;, ² for all A; B 2 U, A + B 2 U, and ² for all ® 2 IR and A 2 U, ®A 2 U. As an example, consider any set of matrices A1;:::; Ak 2 M(m; n; IR). Then Xk hA1;:::; Aki = f ®iAi j ®1; : : : ; ®k 2 IRg i=1 is a subspace of M(m; n; IR). We will be mostly interested in subspaces generated by vectors, i.e., n subspaces of the form U = ha1;:::; aki with ai 2 IR for every i. We say that a1;:::; ak form a T basis of U if they are linearly independent, i.e., ai?aj resp. ai ¢aj = 0 for all i 6= j. In this case, the dimension of a subspace U is equal to k. Two subspaces U1 and U2 are linearly independent, or U1?U2, if and only if any two vectors a 2 U1 and b 2 U2 are linearly independent. Matrices and vectors can also be de¯ned over the complex domain. The conjugate transpose Ay of a matrix A 2 M(m; n; IC) is de¯ned by taking the transpose of A and conjugating all of y its entries. A matrix A 2 M(n; n; IC) is called unitary if and only if A ¢ A = In. 3 n The square norm of a vector a = (ai) 2 IC is de¯ned as Xn 2 2 jjajj = jjaijj i=1 Similar to the real numbers, given any two vectors a; b 2 ICn, the value (ay ¢ b)=jjajj 2 IC is equal to the complex value of the projection of b onto a. Dirac notation In the quantum physics literature, people often use the Dirac notation, or bracket notation, when dealing with vectors. Given a vector a 2 ICn, the bra of a is denoted as haj and represents ay (i.e., the conjugate transpose of a) whereas the ket of a is denoted as jai and simply represents a. Finally, the bracket of a is equivalent to its square norm, namely, hajai = haj ¢ jai = ay ¢ a = jjajj2 : For all vectors a; b; c 2 ICn it holds: ² hajbi = hbjai¤, ² haj®bi = ®hajbi for all ® 2 IC, ² ha + bjci = hajci + hbjci, and ² hajb + ci = hajbi + hajci.