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The mysterious world of quantum computing ____________________________________________ Rajendra K. Bera, PhD Honorary Professor, International Institute of Information Technology, Bangalore Chief Mentor, Acadinnet Education Services India Pvt. Ltd., Bangalore IEEE Workshop on Modern Computing Trends Basaveshwar Engineering College, Bagalkot October 16, 2011

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2 Classical mechanics _________________________________________ A perplexing aspect of quantum mechanics is that it defies an intuitive understanding. It is so different from classical physics as built by Newton, Maxwell, and Einstein. Laws of classical physics are deterministic in the sense that given, say, Newton’s laws of motion, and initial conditions (position and momentum) at some instant t = 0 for a system and a time history of the force(s) acting on the system, we can, in principle, accurately predict the state of that system at any time in the past or the future. In principle, at least, we can measure the state of the system (position and momentum) without disturbing it.

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3 Quantum mechanics _________________________________________ In quantum mechanics, the situation is completely different. The counter- part of Newton’s laws of motion for a quantum system is the Schrödinger’s equation, and the state of the system is described by something called the “wave function”, , which no one understands intuitively. It is so abstract that we understand it only in a mathematical sense. It has not been possible for physicists (or anyone else for that matter) to understand the wave function in any other way. If we try to measure the state of a quantum system, hell breaks loose; we have no way of deter- ministically predicting what the result of a measurement will be! And, even in principle, there is no way we can measure a quantum system without disturbing it. That is why physics is divided into two parts: classical physics, and quantum physics.

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4 Quantum measurement is a mystery _________________________________________ No one knows what transitory changes a quantum system undergoes when it is measured. We do know, however, that while we cannot make a deterministic prediction of the result of a measurement, we can make an amazingly accurate probabilistic prediction of it. I and a former student of mine, Vikram Menon, have come up with a hypothesis to explain this very unusual aspect of quantum systems. You can look up our paper at arXiv: Bera, R.K., and Menon, V., A new interpretation of superposition, entanglement, and measurement in quantum mechanics, arXiv:0908.0957v1 [quant-ph], 07 August 2009, at http://arxiv.org/abs/0908.0957.http://arxiv.org/abs/0908.0957 The probabilistic aspect of quantum mechanics is intriguing because Schrödinger’s equation has no built in probabilities; indeed it produces only deterministic results! So where did the probabilities come in?

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5 Measurement is probabilistic _________________________________________ The probabilities came in because a bunch of physicists, sometime in the 1920s, said so! (This became known as the Copenhagen interpretation of quantum mechanics.) They looked at available experimental data, and they found that the results of measurements carried out on quantum systems follow an unusual probabilistic pattern. Just as Isaac Newton observed that material things are gravitationally attracted to each other (but only “God” knows why) and stated it as a fundamental law of nature, so did Max Born * state this probabilistic aspect of quantum systems as a law of quantum mechanics. It is extremely important to note that laws of nature are like the man-made axioms in mathematics. We do not know (and can never know) why the laws are as they are. Only “God” can enlighten you. We can only marvel at the intellectual genius of those physicists who are able to read the mind of “God”. * Born shared the Nobel Prize in Physics, 1954 (with Walther Bothe) “for his fundamental research in quantum mechanics, especially for his statistical interpretation of the wavefunction”.

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6 Axioms of quantum mechanics _________________________________________ Quantum mechanics describes a physical system through a mathematical object called the state vector (or the wave-function) | . | is complex (i.e., it has real and imaginary parts) and a vector of unit length. | evolves in a deterministic manner according to the linear Schrödinger equation: Here are the laws (or postulates or axioms) of quantum mechanics stated informally. | remains a unit vector during its evolution, only its orientation changes.

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7 Axioms of quantum mechanics (contd) _________________________________________ Any measurement made on a quantum system leads to the (non-unitary) irreversible collapse of its wave function to a new state described by a probability rule. The state space of a composite quantum system is the tensor product of the state spaces of the component quantum systems. It is only when measurements are made that indetermination and probabilities come into quantum theory. Otherwise, things are very deterministic. The collapse of the wave function involves no forces of any kind but it does involve loss of information. Warning: Unless you understand linear algebra and complex variable theory, you will not understand quantum mechanics.

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8 Axioms of quantum mechanics (contd) _________________________________________ The axioms mean, e.g., that the wave function of a system, say, with two possible states |F and |G can be described by the linear combination | = a|F + b|G where a and b are complex constants. After the ‘measurement’, either | = |F or | = |G and these alternatives occur with certain probabilities as noted below: probability |F : probability |G = |a| 2 : |b| 2, |a| 2 + |b| 2 = 1. The collapse of the wave function seems to happen instantly unlike the ordinary time evolution of quantum states (according to the Schrödinger’s equation). We still do not understand the physical mechanism, which causes the collapse. Complex linear superposition of states and collapse of wave functions are unusual features of quantum mechanics.

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9 Axioms of quantum mechanics (contd) _________________________________________ If | has n possible states | 1 , | 2 , | n , it can be described at any instant in time by some unique linear combination as | = a 1 | 1 + a 2 | 2 + + a n | n where a 1, a 2, , a n, are complex constants, which may change with time. After a ‘measurement’, | will collapse to | = | i with the index i occurring with certain probabilities as noted below: prob | 1 : prob | 2 : : prob | n = |a 1 | 2 : | |a 2 | 2 : : |a n | 2, |a 1 | 2 + |a 2 | 2 + + |a n | 2 = 1. When | changes, either due to Schrödinger evolution or measurement, only the a i s change such that the sum of the |a i | 2 remains 1. | therefore remains a unit vector, and only its orientation changes.

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10 Weirdness of quantum mechanics _________________________________________ My wife and my mother-in-law. (W.E. Hill (1915)) http://www.at-ristol.org.uk/Optical/Wife_main.htm What did you see? Question: Just before you realized what the picture represents, was your brain in a state of superposition? Were you in two minds? Hopefully, you now have some understanding of what one means by superposition and wave- function collapse in quantum mechanics. Here is an imperfect example of how weird things can be.

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11 Unitary transformation in QM _________________________________________ The evolution of the wave function, alternatively, can be described in matrix form due to Heisenberg, which is the form used in quantum computing. In this form, the evolution of a closed quantum system is described by a unitary transformation. That is, the state | (t 1 ) of the system at time t 1 is related to the state | (t 2 ) of the system at time t 2 by a unitary operator U which depends only on the times t 1 and t 2, | (t 2 ) = U | (t 1 ) . A linear operator U whose inverse is its adjoint (conjugate transpose, U † ) is called unitary, that is, U † U = UU † = I, where U † (U * ) T (U T ) *. By definition, unitary operators are invertible. Also, by definition, a unitary operator does not change the length of the state vector it acts upon; it only changes that vector’s orientation. This means that if | (t 1 ) = a 1 | a + b 1 | b with |a 1 | 2 + |b 1 | 2 = 1 then | (t 2 ) = a 2 | a + b 2 | b with |a 2 | 2 + |b 2 | 2 = 1.

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12 Quantum computing _________________________________________ Quantum computing is about computing with quantum systems using the rules of quantum mechanics rather than the rules of classical mechanics. The important quantum mechanical phenomena that come into play in the building of a quantum computer are: Superposition Entanglement Decoherence On a quantum computer, programs are executed by unitary evolution of an input that is given by the state of the system. Since all unitary operators are invertible, we can always reverse or ‘uncompute’ a computation on a quantum computer. (We can do this using classical physics also. See: Bennett, C. H., The Thermodynamics of Computation – a Review, International Journal of Theoretical Physics, Vol. 21, No. 12, 1982, pp. 905-940.)

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13 Quantum measurement _________________________________________ In Slide 10, you got a glimpse of the weirdness of QM. I shall now elaborate. Quantum mechanics differs from classical mechanics in the way we interpret the mathematical results. This interpretation is intimately tied with the law related to the measurement of quantum systems. That law is probabilistic in nature. It says that when a quantum system is measured, the wave function | , in general, collapses to a new state according to a probabilistic rule. That is, if | = a|F + b|G then after the ‘measurement’, either | = |F or | = |G and these alternatives occur with certain probabilities. probability |F : probability |G = |a| 2 : |b| 2. ( |a| 2 + |b| 2 = 1 ). A quantum measurement never produces | = a|F + b|G ; a classical measurement does, given an appropriate measuring system.

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14 Quantum superposition _________________________________________ Superposition: Let us look at the picture “My wife, and my mother-in-law”. You cannot really define a linear combination (superposition) of | = a|F + b|G = a|my wife + b|my mother-in-law , in a physical sense, yet you know that in some “complex” sense the two people are superposed in the picture (i.e., they exist simultaneously at the same place and time). You recognize the superposition at an intellectual level, but not at the measurement level (vision); you see the picture “collapsing” to one or the other person. In quantum mechanics, the measurement operator is like a prism—it splits the wavefunction into its component parts (akin to white light being split into its rainbow components by a prism). In our example, the “prism” would split the picture into |my wife and |my mother-in-law and the probability with which we will see one or the other is given by Prob. of seeing |my wife : prob. of seeing |my mother-in-law = |a| 2 : |b| 2. ( |a| 2 + |b| 2 = 1 ). You will not see some weird hybrid form of “my wife” and “my mother-in-law”.

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15 Quantum entanglement _________________________________________ Entanglement: This is a strange state of being in which two particles are so deeply connected that they share the same existence, even when light years apart. Indeed, distance has no meaning for entangled particles. If the state of one is changed, the state of the other is instantly adjusted to be consistent with quantum mechanical rules. If a measurement is made on one, the other will automatically collapse. Entanglement is a form of quantum superposition. There is no easy explanation of entangled correlations. There is no counterpart of entanglement in classical mechanics. Entanglement is a joint characteristic of two or more quantum particles. Einstein called such action-at-a-distance ‘spooky’. [Quote taken from http://www.science-frontiers.com/sf114/sf114p12.htm, William R. Corliss, Science Frontiers #114, Nov-Dec 1997.]http://www.science-frontiers.com/sf114/sf114p12.htm “I cannot seriously believe in [the quantum theory] because it cannot be reconciled with the idea that physics should represent a reality in time and space, free from spooky actions at a distance.” So wrote Einstein to Max Born in March 1947.

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16 Quantum decoherence _________________________________________ Decoherence: It is the spontaneous interaction between a quantum system and its environment. This interaction destroys quantum superposition. The reason why quantum computers still have a long way to go beyond laboratory experimentation is that superposition and entanglement are extremely fragile states. Any interaction with the environment and the particles decohere. Preventing decoherence from taking hold before a calculation is completed remains the biggest challenge in building quantum computers.

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17 Physical laws are mathematical _________________________________________ When we say F = ma expresses Newton’s second law of motion, what we mean is that if you interpret F as representing a force (a vector with 3-scalar components), m representing the mass of a material body, and a representing the acceleration of that material body, then we can very accurately compute the motion of that material body. F = ma means nothing until we give it an interpretation. Surprisingly, all the important laws of physics can be precisely stated in mathematical form. This fact led the 1963 Nobel Laureate in physics, Eugene Paul Wigner, to comment in wonder, The Unreasonable Effectiveness of Mathematics in the Natural Sciences. * It turns out that without knowing mathematics, you cannot develop a deep understanding of physics. This is particularly true for quantum mechanics. * Eugene P. Wigner, The Unreasonable Effectiveness of Mathematics in the Natural Sciences, Communications in Pure and Applied Mathematics, Vol. 13, No. 1 (February 1960), available at http://www.physik.uni- wuerzburg.de/fileadmin/tp3/QM/wigner.pdf.http://www.physik.uni- wuerzburg.de/fileadmin/tp3/QM/wigner.pdf

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18 Interpretation of Schrödinger’s equation _________________________________________ Recall Schrödinger’s equation In quantum mechanics we really do not know what the wave function means. Yet, in Schrödinger’s equation, when we say, that is, make the following interpretation: we make a connection with the physical world. Note that we are interpreting two mathematical operators as physical variables: the operator i is interpreted to represent the physical variable momentum p and the operator i ( / t) as the physical variable energy E! Now you can understand why quantum mechanics is so difficult to understand. It requires a lot of imagination to connect the real world with mathematics.

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19 Why quantum systems are quantized _________________________________________ We now have an interesting situation. Let Then Schrödinger’s equation becomes, in new symbols: Since the Hamiltonian operator can, alternatively, be written as a square matrix, we have an equation of the form we know as an eigenvalue problem in linear algebra: Ax = x. This equation has the trivial solution, x = 0, but it also has non-trivial solutions for certain discrete values of (which we call the eigenvalues of A). So now you see why, for a finite quantum system, energy is quantized. The E values are the eigenvalues of the Hamiltonian operator! A quantum system is quantized because the Schrödinger’s equation describes an eigenvalue problem.

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20 Heisenberg’s uncertainty principle _________________________________________ We now find another intriguing aspect of quantum mechanics. This comes from the fact that operators, do not always commute. Thus, pr rp, because This means that the position r and momentum p that you measure of a particle will depend on the sequence in which you measure them. This is the reason why you cannot measure both position and momentum of a quantum particle with absolute accuracy. And this fact is responsible for the famous Heisenberg’s uncertainty principle in quantum mechanics, which says: p q ħ/2, where q is the error in the measurement of any coordinate and p is the error in its canonically conjugate momentum. In quantum mechanics, position and momentum of a particle are complementary variables.

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21 Heisenberg’s uncertainty principle (contd) _________________________________________ The deep significance of the uncertainty principle is that we cannot observe a quantum system without changing it. The independent observer, watching from the sidelines without influencing the observed phenomenon, simply does not exist. The uncertainty principle essentially says that even in principle it is not possible to know enough about the present to make a complete prediction about the future. This is not so in Newtonian mechanics. The measurement limits imposed by the uncertainty principle cannot be overcome by refining measurement technology; it is a limit imposed by Nature as we penetrate into the subatomic world. Classical and quantum particles are entirely different entities. The principle sets limits on precision technologies, e.g., metrology and lithography. However,

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22 Heisenberg’s uncertainty principle (contd) _________________________________________ if the particle is prepared entangled with a quantum memory (such as an optical delay line) and the observer has access to the particle stored in the quantum memory, it is possible to predict the outcomes for both measurement choices precisely. 1 This is a more general uncertainty relation, formulated in terms of entropies. This new relation has recently been verified experimentally. 2,3 1 M. Berta, et al, The uncertainty principle in the presence of quantum memory, Nature Physics, Vol. 6, Issue 9, September 2010, pp. 659-662. 2 Shuan-Feng Li, et al, Experimental investigation of the entanglement-assisted entropic uncertainty principle, Nature Physics, Vol. 7, Issue 10, October 2011, pp. 752-756. 3 Robert Prevedel, et al, Experimental investigation of the uncertainty principle in the presence of quantum memory and its application to witnessing entanglement, Nature Physics, Vol. 7, Issue 10, October 2011, pp. 757-761.

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23 No cloning, no deletion _________________________________________ That quantum operators are unitary, presents another unusual consequence. One is that if you do not know the state of a quantum system (even if it is a single particle) then you cannot make an exact copy of it. This is known as the no-cloning theorem 1. (You can, of course, prepare a particle in any desired state and make as many copies of it as you like.) The other is that unless a quantum system collapses, you cannot delete information in a quantum system. This is known as the no-deletion theorem 2. These results are connected with the quantum phenomenon called entanglement. Quantum algorithms make very clever use of quantum superposition and quantum entanglement. ______________________ 1 Wootters, W. K., and Zurek, W. H., A single quantum cannot be cloned, Nature, Vol. 299, 28 October 1982, pp. 802-3, http://pm1.bu.edu/~tt/qcl/pdf/wootterw198276687665.pdf. http://pm1.bu.edu/~tt/qcl/pdf/wootterw198276687665.pdf 2 Pati, A. K., and Braunstein, S. L., Impossibility of deleting an unknown quantum state, Nature, Vol. 404, 9 March 2000, pp. 164-5, http://www-users.cs.york.ac.uk/~schmuel/papers/pb00.pdf.http://www-users.cs.york.ac.uk/~schmuel/papers/pb00.pdf

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24 Ingredients of quantum algorithms _________________________________________ Let me now tell you something about quantum algorithms. All quantum algorithms make clever use of choosing and sequencing unitary operators (i.e., moving the system according to Schrödinger’s equation), making measurements (i.e., collapsing the system), and for non-trivial quantum algorithms, making very imaginative use of superposition and entanglement. A single qubit (the quantum analogue of the classical bit) is the simplest quantum system we can think of. Mathematically, a qubit is described as a unit vector | = a|0 + b|1 , parameterized by two complex numbers a and b, satisfying |a| 2 + |b| 2 = 1. While the qubit can be in either state |0 or state |1 (analogous to the 0-1 states of a classical bit), it can also be in a superposed state of a|0 + b|1 (which a classical bit can never be in).

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25 Manipulating single qubits _________________________________________ Any unitary operator M used to manipu- late the state of a qubit can be represented by a linear combination of 4 unitary opera- tors (I, X, Y, Z) ( 0, 1, 2, 3 ), i.e., M = I + X + Y + Z, where , , , are complex constants. When a measurement on a qubit is made, the state of that qubit will collapse, in a probabilistic manner, to either | = |0 or | = |1 according to the rule: probability |0 : probability |1 = |a| 2 : |b| 2. ( |a| 2 + |b| 2 = 1 ). OperationOperator LabelsPost operation qubit state Identity 0, I|0 |0 ; |1 |1 Negation 1, X|0 |1 ; |1 |0 ZX 2, Y|0 |1 ; |1 |0 Phase shift 3, Z|0 |0 ; |1 |1 Note the very important Hadamard gate below. Hadamard H = (X + Z)/ 2|0 (|0 + |1 )/ 2 |+ |1 (|0 |1 )/ 2 | The operators (I, X, Y, Z) are called Pauli matrices, and in their alternative symbolic form ( 0, 1, 2, 3 ) they are called sigma matrices. The operators are 2 2 matrices. The Hadamard gate is a very important and often used gate.

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26 The amazing H-gate _________________________________________ _________________________________________________ After a qubit in state |0 or |1 has been acted upon by a H gate, the state of the qubit is an equal superposition of |0 and |1 . Thus the qubit goes from a deterministic state to a truly random state, i.e., if the qubit is now measured, we will measure |0 or |1 with equal probability. We see that H is its own inverse, that is, H 1 = H or H 2 = I. Therefore, by applying H twice to a qubit we change nothing. This is amazing. By applying a randomizing operation to a random state produces a deterministic outcome! The action of H gate on a qubit is such that |0 (|0 + |1 )/ 2 |1 (|0 |1 )/ 2 and (|0 + |1 )/ 2 |0 (|0 |1 )/ 2 |1

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27 The C not -gate _________________________________________ The C not gate acts on a qubit-pair such that |00 |00 , |01 |01 , |10 |11 , |11 |10 Note that the state of the first qubit, called the control qubit, does not change while the state of the second qubit, called the target qubit, changes only if the control qubit is in state |1 . This signifies the exclusive-or (XOR) operation (that is, the output is “true” if and only if exactly one of the operands has a value of “true”). In quantum computing we rely on the following facts: (1) All quantum gates are reversible. (2) Any unitary operation on n qubits can be implemented exactly by stringing together operations composed of 1-qubit Pauli operators and 2- qubit controlled-NOT gates.

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28 Entangling qubits _________________________________________ An application of the H-gate on the first qubit followed by the C not -gate to a 2-qubit system (with the first qubit as the control qubit) gives the following results when the system’s initial state is |00 , |01 , |10 , and |11 , respectively: C not (H I ) (|00 ) = C not (|00 + |10 )/√2 = (|00 + |11 )/√2 C not (H I ) (|01 ) = C not (|01 |11 )/√2 = (|01 |10 )/√2 C not (H I ) (|10 ) = C not (|00 + |10 )/√2 = (|00 + |11 )/√2 C not (H I ) (|11 ) = C not (|01 |11 )/√2 = (|01 |10 )/√2 The resulting states (in blue) are called Bell states (after John Bell). They are interesting because they are all entangled states, that is, they cannot be attained by manipulating each qubit using the 1-qubit Pauli operators alone. If the states of the entangled particles are used to encode bits, then the entangled joint state represents what is called an ebit. Its state is always distributed between two qubits. The states of these qubits are correlated, but undetermined until measured.

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29 3-qubit Toffoli gate _________________________________________ The T gate (named after Tommaso Toffoli who invented it) acts on a qubit-triplet such that |000 |000 |100 |100 |001 |001 |101 |101 |010 |010 |110 |111 |011 |011 |111 |110 It can be viewed as a controlled-controlled-NOT gate, which negates the last of three bits, if and only if the first two are 1. The Toffoli gate is its own inverse. Toffoli gates can be constructed using six C not gates and several 1-qubit gates. * * See, e.g., Vivek V. Shinde and Igor L. Markov, On the CNOT-cost of TOFFOLI gates, 15 March 2008, arXiv, available at http://arxiv.org/abs/0803.2316; also as Quant. Inf. Comp. 9(5-6):461-486 (2009). http://arxiv.org/abs/0803.2316

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30 Swapping algorithm _________________________________________ The state of two qubits can be swapped by applying the C not gate thrice. Examples |01 |01 with the first qubit as control |11 with the second qubit as control |10 with the first qubit as control |10 |11 with the first qubit as control |01 with the second qubit as control |01 with the first qubit as control |11 |10 with the first qubit as control |10 with the second qubit as control |11 with the first qubit as control The C not gate has no effect on |00 .

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31 Computing x y (x AND y) _________________________________________ Take 3 qubits, each prepared in state |0 , i.e., | = |000 . The first qubit is the placeholder for x, the second for y, and the third for the result of x y. To create the 4 possible inputs of x and y, apply the Hadamard gate to the first two qubits | 1 = (H H I) |000 = (1/ 2) (|0 + |1 ) (1/ 2) (|0 + |1 ) |0 = (1/2) (|000 + |010 + |100 + |110 ). An application of the Toffoli gate, T, now produces | 2 = T| 1 = (1/2) (|000 + |010 + |100 + |111 ). Notice that the 3-qubit system is put into equal superposition of the four possible results of x y. The result of ANDing the first two qubits appears in the third qubit.

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32 Computing x+y _________________________________________ Take 3 qubits, each prepared in state |0 . Compute x y. So we have | 2 = (1/2) (|000 + |010 + |100 + |111 ). Now apply the C not gate to the first two qubits, to get C not I | 2 = (1/2) (|000 + |010 + |110 + |101 ) where the second (blue) qubit is the sum and the third (red) qubit is the carry bit. Note that the carry bit in the adder is the result of an AND operation. The carry and AND are really the same thing. The sum bit comes from an XOR gate (that is, the C not operation).

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33 Interpretations of quantum mechanics ______________________________________________

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34 Interpretations of quantum mechanics _________________________________________ There are several interpretations of quantum mechanics because the wave function is an abstract mathematical object. Neither its origin nor its underlying structure has been disclosed in the laws of quantum mechanics. In particular, the mechanisms for superposition, entanglement, and measure- ment have not been elucidated. Hence, they too are open to interpretation. We shall briefly describe four interpretations—(1) the Copenhagen inter- pretation, (2) Bohm’s interpretation, (3) Everett’s many world interpretation, and (4) Bera- Menon interpretation.

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35 Copenhagen interpretation _________________________________________ In the Copenhagen interpretation (~ 1927), one cannot describe a quantum system independently of a measuring apparatus. Indeed, it is meaningless to ask about the state of the system in the absence of a classical measuring system. The role of the observer is central since it is the observer who decides what he wants to measure. In this interpretation, a particle’s position is essentially meaningless; measurement causes a collapse of the wave function and the collapsed state is randomly picked to be one of the many possibilities allowed for by the system’s wave function; the fundamental objects handled by the equations of quantum mechanics are not actual particles that have an extrinsic reality but “probability waves” that merely have the capability of becoming “real” when an observer makes a measurement. Entangle- ment is treated as a mysterious phenomenon. The Copenhagen interpretation is also known as the ‘shut up and calculate’ inter- pretation (the phrase is due to David Mermin) *. * See Mermin, N. D., Could Feynman have said this? Physics Today, May 2004, pp. 10-11, http://www.physicstoday.org/resource/1/phtoad/v57/i5/p10_s1. http://www.physicstoday.org/resource/1/phtoad/v57/i5/p10_s1

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36 Bohm’s interpretation _________________________________________ In Bohm’s interpretation (1952), the whole universe is entangled; its parts cannot be separated. Entanglement is not a mystery; it is mediated by a very special unknown anti-relativistic quantum information field (pilot wave, derivable from Schrödinger’s equation) that does not diminish with distance and that binds the whole universe together. It is an all pervasive field that is instantaneous; it is not physically measurable but manifests itself in terms of non-local (unmediated, instantaneous, and unaffected by the nature of the intervening medium) correlations. In this interpretation, an electron, e.g., has a well-defined position and momentum at any instant. However, the path an electron follows is guided by the interaction of its own pilot wave with the pilot waves of other entities in the universe. In fact, Bohm treats measurement as an objective process in which the measuring apparatus and what is observed interact in a well-defined way. At the conclusion of this interaction, the quantum system enters into one of a set of ‘channels’, each of which corresponds to the possible results of the measurement while the other channels become inoperative. In particular, there is no ‘collapse’ of the wave function, yet the wave function behaves as if it had collapsed to one of the channels.

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37 Everett’s many world interpretation _________________________________________ Everett’s interpretation (1956) is perhaps the most bizarre and yet perhaps the simplest (it is free of the measurement problem because Everett omits the measure- ment postulate) and, instead, requires us to believe that we inhabit one of an infinite number of parallel worlds! He assumes that when a quantum system in a given world is faced with a choice, i.e., choosing between alternatives as in a measurement, the system splits into a number of systems (worlds) equal to the number of options available. Thus, the world of a qubit in state (|0 + |1 )/ 2 will split into two worlds if the qubit is measured. The two worlds will be identical to each other except for the different option chosen by the qubit—in one it will be in state |0 and in the other it will be in state |1 . Each world will also carry its own copy of the observer(s), and each observer copy will see the specific outcome that appears in his respective world. Of course, the worlds can overlap and interact in the overlapping regions. Decoherence, that is, (spontaneous) interactions between a quantum system and its environment will cause the worlds to separate into non-interacting worlds.

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38 Non-uniqueness of interpretations _________________________________________ What we find in the various interpretations is that while the formalism of quantum mechanics is widely accepted, there is no single interpretation of it that is agreeable to everyone. The disagreements essentially stem from the incompatibility that exists between two evolutionary paths a quantum system follows—the Schrödinger’s equation, and the “collapse” mode of measurement. Indeed, without the measurement postulate telling us what we can observe, the equations of quantum mechanics would be just pure mathematics that would have no physical meaning at all. Note also that any interpretation can come only after an investigation of the logical structure of the postulates of quantum mechanics is made. Let me explain what we mean by an interpretation in the context of quantum mechanics.

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39 Form and meaning are separate _________________________________________ For example, Newtonian mechanics does not define the structure of matter. How we interpret or model the structure of matter is largely an issue separate from Newtonian mechanics. However, any model of the structure of matter we propose is expected to be such that it is compatible with Newton’s laws of motion in the realm where Newtonian mechanics rules. If it is not, then Newtonian mechanics as we know it would have to be abandoned or modified or the model of the structure of matter would have to be abandoned or modified. One may also have a partial interpretation and leave the rest in abeyance till further insight strikes us and leads us to a complete or a new interpretation. A question such as whether a particular result deduced from Newton’s laws of motion is deducible from a given model of material structure is therefore not relevant.

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40 Form and meaning are separate (contd) _________________________________________ Likewise, as long as an interpretation (or model) of superposition, entanglement, and measurement does not require the axioms of quantum mechanics to be altered, none of the predictions made by quantum mechanics would be incompatible with that interpretation. This assertion is important because in our (Bera-Menon) interpretation we make no comments on the Hamiltonian (in the Schrödinger’s equation), which captures the detailed dynamics of a quantum system. Quantum mechanics does not tell us how to construct the Hamiltonian. In fact, real life problems seeking solutions in quantum mechanics need to be addressed in detail by physical theories built within the framework of quantum mechanics. The postulates of quantum mechanics provide only the scaffolding around which detailed physical theories are to be built.

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41 A new interpretation (Bera-Menon, 2009) _________________________________________ In our interpretation, we provide a sub-Planck-scale view of the wave function, superposition, entanglement, and measurement without affecting the postulates of quantum mechanics. The sub-Planck scale is chosen to provide us with the freedom to construct mechanisms for our interpretation that are not necessarily bound by the laws of quantum mechanics (just as atomic structure is not bound by Newtonian mechanics). In particular our interpretation does not have to satisfy the Schrödinger wave equation because quantum mechanics is not expected to rule in the sub-Planck scale. The high point of our interpretation is that it is able to explain the measure- ment postulate as the inability of a classical measuring device to measure at a precisely predefined time. Ref. Bera, R.K., and Menon, V., A new interpretation of superposition, entanglement, and measurement in quantum mechanics, arXiv:0908.0957v1 [quant-ph], 07 August 2009, at http://arxiv.org/abs/0908.0957.http://arxiv.org/abs/0908.0957

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42 A new interpretation (Superposition) _________________________________________ That is, the superposed states appear as time-sliced in a cyclic manner such that the time spent by an eigenstate in a cycle is related to the complex amplitudes (a, b) appearing in the qubit’s wave function, | = a |0 + b |1 . The cycle time T c of the qubit’s oscillation between states |0 and |1 is much smaller than the Planck time (<< 10 -43 sec). It is not necessary for us to know the value of T c. We only assert that it is a universal constant. Within a cycle, the time spent by the particle in state |0 is T 0 = |a| 2 T c and in state |1 is T 1 = |b| 2 T c so that T c = T 0 + T 1. In our interpretation we assume that the sub-Planck scale structure of the wave function is such that the wave function is in only one state at any instant but oscillates between its various “superposed” component states (eigenstates). (There is no expenditure of energy in maintaining the oscillations.) State |0 State |1 T0T0 T1T1 TcTc

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43 A new interpretation (Measurement) _________________________________________ Our hypothesized measurement mechanism acts instantaneously (through entanglement) but the instant of actual measurement occurs randomly over a small but finite interval t m, which is much greater than Planck time (otherwise its actual value is immaterial), from the time the measurement apparatus is activated. In particular, we regard measurement as the joint product of the quantum system and the macroscopic classical measuring apparatus. To avoid bias, we assume that the device can choose any instant in the interval t m with equal probability. Thus the source of indeterminism built into quantum mechanics is interpreted here as occurring due to the classical measuring device’s inability to measure at a precisely predefined time. We do not explain how the collapse of the wave function occurs when a measurement is made, only why the measurement outcome is prob- abilistic. Once a measurement is made, the wave function assumes the collapsed state.

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44 A new interpretation (Entanglement) _________________________________________ Entangled states binding two or more qubits appear in our interpretation as the synchron- ization of the sub-Planck level oscillation of the participating qubits, as shown below for the two-qubit system Bell states, (|00 ± |11 )/√2 (|01 ± |10 )/√2 A measurement on one of the entangled qubits will collapse both simultaneously to the respective state they are in at the instant of measurement (such as 1 or 2 in the Figure). We do not know how Nature might accomplish the required synchronization. 1 : |00 2 : |11 Particle 1 Particle 2 TcTc 2 : |10 1 : |01 Particle 1 Particle 2 TcTc

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45 A new interpretation (Entanglement) (contd) _________________________________________ It is, of course, clear that our interpretation cannot violate the uncertainty principle since the latest measurement on a system collapses the system according to the measurement postulate. Thus, there can be no direct correlation between any earlier results of measurement on the system, and the succeeding measurement. Unlike the Copenhagen interpretation, in our interpretation it is not meaningless to ask about the state of the system in the absence of a measuring system.

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46 Thank you! There is no quantum world. There is only an abstract physical description. It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature. — Niels Bohr This view is very different from that of Einstein’s who believed that the job of physical theories is to ‘approximate as closely as possible to the truth of physical reality.’ ________________________________________________________________

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Integrals over Operators

Integrals over Operators

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