Understanding the Link between Quantum Teleportation and Computing - Prof. David Peak, Lab Reports of Physics

Download Understanding the Link between Quantum Teleportation and Computing - Prof. David Peak and more Lab Reports Physics in PDF only on Docsity! Phys2710 “Teleportation” (2007) 1 Quantum Teleportation and Quantum Computing “Teleportation … is the process of moving objects from one place to another by encoding information about the object, transmitting the information to another place, … and creating a copy of the original object in the new location. The use of teleportation has traditionally been found only in science fiction, but the theory and experimentation of quantum teleportation has [recently] been of interest to physicists.” (From a former posting on Wikipedia.) The definition given above actually has been realized in several experiments that use the property of entanglement (see “Spin and the Spookiness of Quantum Reality”) to perform something called quantum teleportation (QT). QT has been achieved over a distance of 600 m, but so far, at least, has not come very close to producing what happens in Star Trek’s transporter, but might (perhaps in 2139—the putative date of invention of the transporter) eventually get there. The way QT works is shown schematically in the diagram to the right. E1 and E2 are two particles whose states are entangled. In this case, that means the state of E1 can be completely determined by measuring the state of E2. A is an entity about which nothing is known initially. The object is to teleport A (that is, make an identical copy of A) from the left hand side of the diagram to the right hand side—perhaps far away. E1 and A interact—that’s depicted in the first gray ball in the diagram. The interaction destroys the initial states of E1 and A, but the outgoing state of E1 is measured and that information is sent (via the “dotted line” in the diagram) to E2 (over an ordinary communication link, like a fax or something). When the state of E2 is measured that tells us what the initial state of E1 must have been. Since the message gives us the state of E1 after the interaction it contains information about what kind of particle A is and what its initial state must have been. Thus, on the right side of the diagram we can retrieve an A-type entity (A′) from storage and coerce it to be in the same state as A was initially, producing A′′—an exact copy of the original A. It’s as if A has been transported from the left to the right without actually having traveled in between. The eventual applications of QT (if any) are still unknown, but the phenomenon certainly harbors interesting potential. Quantum computing (QC), also exploits entanglement. QC holds the potential to become the computing paradigm of the future—at least, for solving some really hard problems. In a classical computer, computation involves retrieving bit strings (sequences of 1s and 0s) from memory registers, passing them through suites of logic gates, and returning the new strings to memory. This is often physically realized by manipulating the voltage on microscopic circuit elements. An example of a classical computation step is what happens at an XOR gate. The gate accepts input bits at nodes P and Q and returns a bit at the node P_XOR_Q according to the table P Q P_XOR_Q 0 0 0 1 0 1 0 1 1 1 1 0 The XOR gate has the effect of reducing the bit string from 2 bits to 1, and because there are two output 0s and two output 1s, which input state produces which output is ambiguous. The classical XOR gate is therefore said to be irreversible. Most of the logic gates of classical computers are irreversible, so if only the output bits produced by a series gates are known, it is not possible to reconstruct the input bits. In addition, when a gate reduces the number of known bits from two to one, it reduces the entropy of the computer. But Nature doesn’t allow entropy to be reduced so entropy has to increase elsewhere. This Phys2710 “Teleportation” (2007) 2 entropy increase shows up as heat released into the environment. Because of irreversibility, a classical computer has to “run hot.” In quantum computation, on the other hand, bits are coded in terms of quantum variables, such as spin states. Thus, an up spin (↑) might be taken to represent a 1 and a down spin (↓) a 0; such spin- encoded bits are called Q-bits (for “quantum bits”). In quantum computation, a logic gate corresponds to a quantum operator that transforms Q-bits. Suppose that an operator can be manufactured that transforms the spins of two input particles into two new spins as follows: Spins in Spins out ↓↓ ↓↓ ↑↓ ↑↑ ↓↑ ↓↑ ↑↑ ↑↓ Note that the transformations shown only affect the rightmost of the two spins; the leftmost spin is unaltered. This set of transformations can be interpreted as an XOR gate provided that the leftmost spin of the input state is taken to be the “P” input Q-bit, the rightmost spin of the input state is taken to be the “Q” input Q-bit, and the rightmost spin of the output state is taken to be the “P_XOR_Q” Q-bit. Note that the leftmost spin of the output state can be used to remove the ambiguity about which state has produced which output. In other words, in this example of a quantum XOR gate, the computation is totally reversible: reading both spins in the output state tells you what the input state was—provided, of course, that no interaction other than the XOR operator occurred to the two spins before the output was read. As long as the latter is true, the input and output states are strongly correlated: they are entangled. If the input and output states remain entangled it is always possible to encode inputs and outputs somehow so that the output state can unambiguously tell what the input state was. Because of this, all quantum logic gates are all designed to be reversible. Passing information through a quantum gate produces no change in entropy, so during the processing quantum computers generate no heat. One could imagine initiating a quantum computation with a superposition of inputs, such as ↓↓+↑↓+↓↑+↑↑. The XOR operator transforms this input superposition state into the superposition ↓↓+↑↑+↓↑+↑↓. It is tempting to declare that the XOR operator has taken all possible inputs and computed all possible outputs in one operation, thus reducing by a factor of four how XOR would be computed classically, one input at a time. But that’s not right. From the point of view of quantum reality there is no computation until “someone looks.” Until then there is only the possibility of a quantum computation. “Looking” means performing a measurement, the result of which will yield at random either ↓↓ or ↑↑ or ↓↑ or ↑↓—one single output each time the measurement is made. To deduce what an XOR gate does classically, you observe the output for each possible input, i.e., four iterations. In a quantum computer, if you were really unlucky you might get the same ↑↑ 100 times in a row. That’s WAY worse than a classical computer. Moreover, the apparent saving of energy is a misconception also, because when you look you collapse the input state from 8 bits to 2, a four-fold reduction in entropy as opposed to the two-fold reduction in a classical XOR. That is, when you look, you get twice as much heat out of a quantum XOR. So what’s a quantum computer good for, then? Here’s an example, due to David Mermin (Quantum Computer Science, Cambridge, 2007). Suppose the function f(x) maps the bit x (i.e., 0 or 1) into the millionth bit of ! 2+ x (i.e., ! 2 or ! 3 ). Suppose you want to know, does ! f (0) = f (1) (i.e., does the millionth bit of ! 2 = the millionth bit of ! 3 )? Using a classical computer you would evaluate both f(0) and f(1) and compare the results: two iterations. But, a fiendishly clever quantum algorithm has been found that does the following. Assume that a state is a Q-bit pair (xy). In this algorithm, a series of gates (transformations) takes the input state (00) into (0w) if ! f (0) = f (1) and (1z) otherwise. In the output states, w and z are Q-bits neither one of which can be used to evaluate f. In other words, by measuring the