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**University of Queensland**

Quantum Computation Michael A. Nielsen University of Queensland Goals: To explain the quantum circuit model of computation. To explain Deutsch’s algorithm. To explain an alternate model of quantum computation based upon measurement.

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**What does it mean to compute?**

Church-Turing thesis: An algorithmic process or computation is what we can do on a Turing machine. Deutsch (1985): Can we justify C-T thesis using laws of physics? Quantum mechanics seems to be very hard to simulate on a classical computer. Might it be that computers exploiting quantum mechanics are not efficiently simulatable on a Turing machine? Violation of strong C-T thesis! Might it be that such a computer can solve some problems faster than a probabilistic Turing machine? Candidate universal computer: quantum computer

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**The Church-Turing-Deutsch principle**

Church-Turing-Deutsch principle: Any physical process can be efficiently simulated on a quantum computer. Research problem: Derive (or refute) the Church- Turing-Deutsch principle, starting from the laws of physics.

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**Models of quantum computation**

There are many models of quantum computation. Historically, the first was the quantum Turing machine, based on classical Turing machines. A more convenient model is the quantum circuit model. The quantum circuit model is mathematically equivalent to the quantum Turing machine model, but, so far, human intuition has worked better in the quantum circuit model. There are also many other interesting alternate models of quantum computation!

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**Quantum circuit model Quantum Classical Unit: bit Unit: qubit**

1. Prepare n-bit input Prepare n-qubit input in the computational basis. and 2-bit logic gates 2. Unitary 1- and 2-qubit quantum logic gates 3. Readout partial information about qubits 3. Readout value of bits External control by a classical computer.

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**Single-qubit quantum logic gates**

Pauli gates Hadamard gate Phase gate =

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**Controlled-phase gate**

Controlled-not gate Control CNOT is the case when U=X Target U Controlled-phase gate Z = Z Z Symmetry makes the controlled-phase gate more natural for implementation! = X H Z H Exercise: Show that HZH = X.

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**Toffoli gate Worked Exercise: Show that all permutation matrices**

Control qubit 1 Control qubit 2 Target qubit Worked Exercise: Show that all permutation matrices are unitary. Use this to show that any classical reversible gate has a corresponding unitary quantum gate. Challenge exercise: Show that the Toffoli gate can be built up from controlled-not and single-qubit gates. Cf. the classical case: it is not possible to build up a Toffoli gate from reversible one- and two-bit gates.

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**How to compute classical functions**

on quantum computers Use the quantum analogue of classical reversible computation. The quantum NAND The quantum fanout Classical circuit Quantum circuit

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**Removing garbage on quantum computers**

Given an “easy to compute” classical function, there is a routine procedure we can go through to translate the classical circuit into a quantum circuit computing the canonical form.

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**Example: Deutsch’s problem**

Classical black box Quantum black box

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**Putting information in the phase**

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**Quantum algorithm for Deutsch’s problem**

Quantum parallelism Research problem: What makes quantum computers powerful?

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**Universality in the quantum circuit model**

Classically, any function f(x) can be computed using just nand and fanout; we say those operations are universal for classical computation. Suppose U is an arbitrary unitary transformation on n qubits. Then U can be composed from controlled-not gates and single-qubit quantum gates. Just as in the classical case, a counting argument can be used to show that there are unitaries U that take exponentially many gates to implement. Research problem: Explicitly construct a class Un of unitary operations taking exponentially many gates to implement.

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**Summary of the quantum circuit model**

QP: The class of decision problems soluble by a quantum circuit of polynomial size, with polynomial classical overhead.

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**Quantum complexity classes**

How does QP compare with P? BQP: The class of decision problems for which there is a polynomial quantum circuit which outputs the correct answer (“yes” or “no”) with probability at least ¾. BPP: The analogous classical complexity class. Research problem: Prove that BQP is strictly larger than BPP. Research problem: What is the relationship of BQP to NP?

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**When will quantum computers be built?**

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**Alternate models for quantum computation**

Standard model: prepare a computational basis state, then do a sequence of one- and two-qubit unitary gates, then measure in the computational basis. Research problem: Find alternate models of quantum computation. Research problem: Study the relative power of the alternate models. Can we find one that is physically realistic and more powerful than the standard model? Research problem: Even if the alternate models are no more powerful than the standard model, can we use them to stimulate new approaches to implementations, to error-correction, to algorithms (“high-level programming languages”), or to quantum computational complexity? The first alternate universal set I’m going to describe today tips this model completely on its head. In this alternate set there are no coherent unitary dynamics whatsoever- it doesn’t even make sense to talk about the time required to perform a unitary gate. Instead, the only coherent operation in the computer is quantum memory – the storage of quantum information. In addition to this, we also require the ability to do 2-qubit projective measurements. As I’ll explain in a moment we require the ability to do these measurements in an arbitrary, possibly entangled basis. Now, sometimes when I explain this result to people, they say “but isn’t the ability to do 2-qubit projective measurements in an arbitrary basis just as hard as doing a quantum computation?” Well, that’s certainly seems to me to be the case. At the present point in time I don’t see that this model is especially useful as a practical device, although maybe time will prove me wrong. However, what does strike me as extremely important is the general insight that far from being an enemy, the measurements that we usually think of as destroying coherence can actually be our friend. Indeed, they are all that’s required to do universal quantum computation. I should mention, by the way, that this set was proved universal by a combination of two papers, a quant-ph I wrote earlier this year, in which I needed four-qubit projective measurements, and a nice recent paper by Debbie Leung. There is another closely related paper that I’m not going to talk about today, although I believe Hans Briegel will later this week, a paper by Raussendorf and Briegel that appeared this year in PRL. What they showed is that it’s actually possible to replace the 2-qubit projective measurements by 1-qubit projective measurements, provided one is supplied with an array of qubits in a particular entangled state. I’m not going to discuss this work today, but clearly the moral is in part the same: measurements are a useful tool in doing quantum computation. A similar moral is evident in the work of Knill, Laflamme and Milburn, where they show that linear optics, plus the ability to prepare states and do measurements in the number state basis, is universal for quantum computation.

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**Alternate models for quantum computation**

Overview: Alternate models for quantum computation Topological quantum computer: One creates pairs of “quasiparticles” in a lattice, moves those pairs around the lattice, and then brings the pair together to annihilate. This results in a unitary operation being implemented on the state of the lattice, an operation that depends only on the topology of the path traversed by the quasiparticles! Quantum computation via entanglement and single- qubit measurements: One first creates a particular, fixed entangled state of a large lattice of qubits. The computation is then performed by doing a sequence of single-qubit measurements. The first alternate universal set I’m going to describe today tips this model completely on its head. In this alternate set there are no coherent unitary dynamics whatsoever- it doesn’t even make sense to talk about the time required to perform a unitary gate. Instead, the only coherent operation in the computer is quantum memory – the storage of quantum information. In addition to this, we also require the ability to do 2-qubit projective measurements. As I’ll explain in a moment we require the ability to do these measurements in an arbitrary, possibly entangled basis. Now, sometimes when I explain this result to people, they say “but isn’t the ability to do 2-qubit projective measurements in an arbitrary basis just as hard as doing a quantum computation?” Well, that’s certainly seems to me to be the case. At the present point in time I don’t see that this model is especially useful as a practical device, although maybe time will prove me wrong. However, what does strike me as extremely important is the general insight that far from being an enemy, the measurements that we usually think of as destroying coherence can actually be our friend. Indeed, they are all that’s required to do universal quantum computation. I should mention, by the way, that this set was proved universal by a combination of two papers, a quant-ph I wrote earlier this year, in which I needed four-qubit projective measurements, and a nice recent paper by Debbie Leung. There is another closely related paper that I’m not going to talk about today, although I believe Hans Briegel will later this week, a paper by Raussendorf and Briegel that appeared this year in PRL. What they showed is that it’s actually possible to replace the 2-qubit projective measurements by 1-qubit projective measurements, provided one is supplied with an array of qubits in a particular entangled state. I’m not going to discuss this work today, but clearly the moral is in part the same: measurements are a useful tool in doing quantum computation. A similar moral is evident in the work of Knill, Laflamme and Milburn, where they show that linear optics, plus the ability to prepare states and do measurements in the number state basis, is universal for quantum computation.

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**Alternate models for quantum computation**

Overview: Alternate models for quantum computation Quantum computation as equation-solving: It can be shown that quantum computation is actually equivalent to counting the number of solutions to certain sets of quadratic equations (modulo 8)! Quantum computation via measurement alone: A quantum computation can be done simply by a sequence of two-qubit measurements. (No unitary dynamics required, except quantum memory!) The first alternate universal set I’m going to describe today tips this model completely on its head. In this alternate set there are no coherent unitary dynamics whatsoever- it doesn’t even make sense to talk about the time required to perform a unitary gate. Instead, the only coherent operation in the computer is quantum memory – the storage of quantum information. In addition to this, we also require the ability to do 2-qubit projective measurements. As I’ll explain in a moment we require the ability to do these measurements in an arbitrary, possibly entangled basis. Now, sometimes when I explain this result to people, they say “but isn’t the ability to do 2-qubit projective measurements in an arbitrary basis just as hard as doing a quantum computation?” Well, that’s certainly seems to me to be the case. At the present point in time I don’t see that this model is especially useful as a practical device, although maybe time will prove me wrong. However, what does strike me as extremely important is the general insight that far from being an enemy, the measurements that we usually think of as destroying coherence can actually be our friend. Indeed, they are all that’s required to do universal quantum computation. I should mention, by the way, that this set was proved universal by a combination of two papers, a quant-ph I wrote earlier this year, in which I needed four-qubit projective measurements, and a nice recent paper by Debbie Leung. There is another closely related paper that I’m not going to talk about today, although I believe Hans Briegel will later this week, a paper by Raussendorf and Briegel that appeared this year in PRL. What they showed is that it’s actually possible to replace the 2-qubit projective measurements by 1-qubit projective measurements, provided one is supplied with an array of qubits in a particular entangled state. I’m not going to discuss this work today, but clearly the moral is in part the same: measurements are a useful tool in doing quantum computation. A similar moral is evident in the work of Knill, Laflamme and Milburn, where they show that linear optics, plus the ability to prepare states and do measurements in the number state basis, is universal for quantum computation. Further reading on the last model: D. W. Leung,

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**Can we build a programmable quantum computer?**

fixed array Challenge exercise: Prove the no-programming theorem.

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**A stochastic programmable quantum computer**

Bell Measurement {

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Why it works Bell Measurement {

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**How to do single-qubit gates using measurements alone**

Bell Measurement {

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Coping with failure What happens the other 3/4 ‘s of the time, when j is not equal to k? Well, we know both j and k, and the product U sigma k sigma j is unitary, so we have a known unitary operation that was applied to the qubit. It’s not the desired effect, but it is a known unitary operation, and we can easily fix it up.

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**How to do the controlled-not**

Bell Measurement {

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**Discussion Measurement is now recognized as a powerful tool in many**

schemes for the implementation of quantum computation. Research problem: Is there a practical variant of this scheme? Research problem: What sets of measurement are sufficient to do universal quantum computation? Research problem: Later in the week I will talk about attempts to quantify the “power” of different entangled states. Can a similar quantitative theory of the power of quantum measurements be developed? To summarize, what I’ve shown is that an arbitrary quantum computation can be efficiently performed using just quantum memory and four-qubit projective measurements. Fenner and Zhang have simplified the scheme I described so that only three-qubit measurements are required, and Debbie Leung has simplified it even further to two. I should state, incidentally, that I was pretty surprised by Debbie’s result – I thought it would actually be impossible to reduce to two-qubit measurements, and spent a lot more time looking for an impossibility proof than I did looking for a measurement scheme. A couple of obvious questions suggest themselves. First, is there a practical variant of this scheme? At the moment, I don’t know of any way of doing it practically. Something I should state is that, so far as I know, the threshold for this scheme is substantially lower than in the regular model. If you assume that the regular threshold is 10 to the minus 5, and then use this scheme to simulate the regular model, you end up with a threshold of about 10 to the minus 6 or 7, which is quite a bit worse. It would be interesting to determine if there are more direct ways of doing fault-tolerant computation that lead to a better threshold for this model. A more fundamental question, to my mind is the question of what sets of measurement are sufficient to do universal quantum computation?

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