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Teleportation As a Quantum Computation

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Teleportation As a Quantum Computation PhysComp96 Teleportation as a Extended abstract Draft, 12 May 1996 quantum computation Gilles Brassard, frsc∗ Universit´ede Montr´eal† 1 Introduction of quantum teleportation would be greatly reduced if the actual teleportation had to take place immediately after Among the many exciting new applications of quantum the required preparation. Thus, a working demonstra- physics in the realm of computation and information tion of quantum teleportation is likely to be seen before theory, I am particularly fond of quantum cryptogra- the quantum factorization of even a very small integer phy, quantum computing and quantum teleportation [5]. is achieved, but quantum teleportation across significant Quantum cryptography allows for the confidential trans- time and space will have to await a technology that allows mission of classical information under the nose of an eaves- for the efficient long-term storage of quantum informa- dropper, regardless of her computing power or technolog- tion. Nevertheless, it may be that short-distance quantum ical sophistication [2, 1, 4]. Quantum computing allows teleportation will play a role in transporting quantum in- for an exponential amount of computation to take place formation inside quantum computers. Thus we see that simultaneously in a single piece of hardware [9, 7]; of par- the fates of quantum computing and quantum teleporta- ticular interest is the ability of quantum computers to tion are entangled! factorize numbers very efficiently [14], with dramatic im- plications for classical cryptography [6]. Quantum tele- portation allows for the transmission of quantum infor- 2 Quantum teleportation mation to a distant location despite the impossibility of measuring or broadcasting the information to be trans- Recall that any attempt at measuring quantum informa- mitted [3]. Each of these concepts had a strong overtone tion disturbs it irreversibly and yields incomplete infor- of science fiction when they were first introduced. mation. This makes it impossible to transmit quantum information through a classical channel. Recall also that If asked to rank these ideas on a scale of technologi- the purpose of quantum teleportation [3] is to circumvent cal difficulty, it is tempting to think that quantum cryp- this impossibility so as to allow the faithful transmission tography is easiest while quantum teleportation is the of quantum information between two parties, convention- most outrageous—especially when it comes to teleport- ally referred to as Alice and Bob. ing goulash [11]! This ranking is correct with respect to In order to achieve teleportation, Alice and Bob must quantum cryptography, whose feasibility has been demon- strated by several experimental prototypes capable of reli- share prior quantum entanglement. This is usually ex- ably transmitting confidential information over distances plained in terms of Einstein–Podolsky–Rosen nonlocal quantum states [8] and Bell measurements, which makes of tens of kilometres [12, 13, 10]. The situation is less clear when it comes to comparing the technological feasibility of the process seem very mysterious. The purpose of this note is to show how to achieve quantum teleportation very quantum computing with that of quantum teleportation. simply in terms of quantum computation. As interesting arXiv:quant-ph/9605035v1 23 May 1996 On the one hand, quantum teleportation can be im- side product, we obtain a quantum circuit with the un- plemented with a quantum circuit that is much simpler usual feature that there are points in the circuit at which than that required by any nontrivial quantum computa- the quantum information can be completely disrupted by tional task: the state of an arbitrary qubit (quantum bit) a measurement—or some types of interaction with the can be teleported with as few as two quantum exclusive- environment—without ill effects: the same final result is or (controlled-not) gates. Thus, quantum teleportation is obtained whether or not measurement takes place. This significantly easier to implement than quantum comput- is true despite that fact that the qubits affected by these ing if we are concerned only with the complexity of the measurements are entangled with the other qubits carried required circuitry. by the circuit, which should make these measurements On the other hand, quantum computing is meaning- even more damaging. ful even if it takes place very quickly—indeed its primary Of course, the uncanny power of quantum computation purpose is increased computational speed—and within a draws in parts on nonlocal effects inherent to quantum small region of space. Quite the opposite, the interest mechanics. The quantum teleportation circuit described in 4 is not really different in principle from the origi- ∗Supported in part by Nserc and Fcar § †D´epartement IRO, C.P. 6128, succursale centre–ville, Montr´eal, nal idea [3] since it uses quantum computation to create Canada H3C 3J7. email: [email protected] and measure nonlocal states. Nevertheless it sheds new 1 α light on teleportation, at least from a pedagogical point if α 0 + β 1 is represented by vector (β ). Similarly the of view, since it makes the process completely straight- quantum| i exclusive-or| i operation is given by matrix forward to anyone who believes that quantum computa- tion is a reasonable proposition. Moreover, this circuit 1 00 0 0 10 0 could genuinely be used for teleportation purposes inside XOR = 0 00 1 a quantum computer. Finally, the surprising resilience of 0 01 0 this circuit to measurements performed while it is pro- cessing information may turn out to have relevance to if α 00 + β 01 + γ 10 + δ 11 is represented by the quantum error correction. transpose| i of| vectori (|α,β,γ,δi |). i 3 The basic ingredients 4 The teleportation circuit As is often the case with quantum computation, we shall Consider the following quantum circuit. Please disregard need two basic ingredients: the exclusive-or gate (also the dashed line for the moment. known as controlled-not), which acts on two qubits at once, and arbitrary unitary operations on single qubits. a t R S i S i x Let 0 and 1 denote basis states for single qubits and recall| i that pure| i states are given by linear combination of basis states such as ψ = α 0 + β 1 where α and β are b L ti t y complex numbers such| i that | αi 2 + |βi2 = 1. The quantum exclusive-or| (XOR),| | | denoted as follows, c i it T t z a t x Alice Bob b i y Let ψ be an arbitrary one-qubit state. Consider what sends 00 to 00 , 01 to 01 , 10 to 11 and 11 to 10 . | i In other| words,i | providedi | i the| i input| i states| i at a and| ib are| ini happens if you feed ψ00 in this circuit, that is if you set upper input a| to iψ and both other inputs b basis states, the output state at x is the same as the input | i state at a, and the output state at y is the exclusive-or and c to 0 . It is a straightforward exercise to verify that state| ψi will be transferred to the lower output z, of the two input states at a and b. This is also known | i as the controlled-not gate because the state carried by whereas both other outputs x and y will come out in the control wire “ax” is not disturbed whereas the state state φ = ( 0 + 1 )/√2. In other words the output will be φφψ| i . If| thei two| i upper outputs are measured in the carried by the controlled wire “by” is flipped if and only | i if the state on the control wire was 1 . Note that the standard basis ( 0 versus 1 ), two random classical bits | i will be obtained| ini addition| i to quantum state ψ on the classical interpretation given above no longer holds if the | i input qubits are not in basis states: it is possible for the lower output. output state on the control wire (at x) to be different from Now, let us consider the state of the system at the its input state (at a). Moreover, the joint state of the dashed line. A simple calculation shows that all three output qubits can be entangled even if the input qubits qubits are entangled. We should therefore be especially were not, and vice versa. careful not to disturb the system at that point. Never- In addition to the quantum exclusive-or, we shall need theless, let us measure the two upper qubits, leaving the two single-qubit rotations L and R, and two single-qubit lower qubit undisturbed. This measurement results in two conditional phase-shifts S and T. Rotation L sends 0 purely random classical bits u and v, bearing no correla- | i tion whatsoever with the original state ψ . Let us now to ( 0 + 1 )/√2 and 1 to ( 0 + 1 )/√2, whereas | i R sends| i 0| ito ( 0 1|)/i√2 and−| i1 to| i ( 0 + 1 )/√2. turn u and v back into quantum bits and reinject u and | i | i − | i | i | i | i v in the circuit immediately after the dashed line.| i Note that LR ψ = RL ψ = ψ for any qubit ψ . Condi- | i tional phase-shift| i S sends| i 0| toi i 0 and leaves| i1 undis- Needless to say that the quantum state carried at turbed, whereas T sends | 0i to | i 0 and 1 |toi i 1 . the dashed line has been completely disrupted by this In terms of unitary matrices,| i the operations−| i | arei − | i measurement-and-resend process. We would therefore ex- pect this disturbance to play havoc with the final output 1 1 1 1 1 1 of the circuit.
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