Conference ELMECO‐6 ELECTROMAGNETIC DEVICES AND PROCESSES IN ENVIRONMENT PROTECTION joint with th 9 Seminar APPLICATIONS OF SUPERCONDUCTORS AoS‐9 June 24 – 27, 2008 Nałęczów, Poland
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INTRODUCTION TO QUANTUM COMPUTING Leszek JAROSZYŃSKI Lublin University of Technology Abstract. It appears that "quantum computers" has already come from science-fiction to the reality. Quantum computing shows outstanding efficiency in some numerical problems. Length of qubit registers grows noticeably last years – at least so fast as the number of interesting quantum algorithms. It's time to manufacture qubit integrated circuits. These days two promising concepts fight their way to a real usage: Josephson junction qubits and quantum dot qubits. Streszczenie. Wydaje się, że tzw. komputery kwantowe przeniknęły już z kart literatury fantastyczno-naukowej do rzeczywistości. Algorytmy kwantowe wykazują znakomitą wydajność w przypadku niektórych problemów. Długość rejestrów qubitowych rośnie każdego roku – co najmniej tak szybko jak liczba interesujących algorytmów kwantowych. Nadszedł czas wytwarzania rejestrów qubitowych w postaci układów scalonych. Aktualnie ścierają się dwie obiecujące koncepcje: rejestrów qubitowych ze złączami Josephsona i kropkami kwantowymi.
Keywords: qubit, quantum computing, Josephson junction. Słowa kluczowe: qubit, obliczenia kwantowe, złącze Josephsona.
Introduction A quantum (plural: quanta) in an indivisible portion of physical quantity. Energy and momentum, same scientists state that also length and time, can take only certain discrete values. The distance between two adjacent levels is a quantum. At quantum dimensions classical physics fails. A new idea – quantum physics – can explain all incredible phenomena. The history of quantum physics is quite long. Everything started with discrepancy between the theory of a black body radiation and experimental results. After a few months of th intellectual work, on 14 of December 1900 Max Planck presented explanation of his own improvement made for the black body emissivity equation. He postulated that the electromagnetic energy could be emitted only in a quantized manner - the energy could only be a multiple of an elementary quantity. That opened a new era in physics [1]. Qubit For practical reasons we are used to do calculations using binary devices. A bit can hold binary portion of information: zero or one and nothing more. A qubit has also two basis states – let's call them ket zero |0> and ket one |1>. However, unlike a bit, qubit can hold a linear superposition of those two states (1). (1)
ψ = a 0 +b1
where: a, b – complex probability amplitudes. The state space of a qubit is usually represented as the surface of a Bloch sphere (Fig. 1). It has two degrees of freedom so when we measure qubit we get state |0> with 2 2 probability |a| and state |1> with probability |b| . Nothing is certain, it's just probable. Single qubit doesn't seem so interesting – only qubit register shows full potential of this idea.
in terms of the states of its qubits. It's called "entangled state". Unlike bit registers, the n-qubit register has a state space of 2n dimensions just because entanglement. Therefore, 270-bit register can hold one very big number and 270-qubit system represents in parallel more states than the number of atoms in the Universe. Unfortunately, there are some problems concerning reading of register states. Any access to a qubit – so called measurement disturbs the quantum state. Moreover, any result is only probabilistic. |0> |ψ>
z θ
y
φ x |1>
Fig. 1. A qubit representation as a Bloch sphere
Algorithms Regardless mentioned problems, in the past few years some excellent ideas for the overcoming of the measurement problem have been found. Additionally, quantum algorithms probably outperform any known classical ones. In the Table 1, a comparison of chosen algorithms has been presented. Table 1. Comparison of quantum and classical algorithms [2] classical quantum name complexity complexity
Quantum entanglement Quantum entanglement is a phenomenon where the quantum states of two or more qubits are bound together. Sometimes the state of quantum register can't be described
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Simon's classification Grover's searching
n
O( 2 2 )
O( n)
O( n )
O( n )
1
Shor's factoring
2
3 3 O(e (log n) (log log n) )
O( n 3 )
Implementations Despite the catchy abstract presented above, the problem of the manufacture of usable quantum computers is still open. There are some tested concepts of building quantum computers using different approaches. They are shown at Table 2. Table 2. Qubit implementations [2, 3] physical information "0" object support photon
coherent light electron
M
"1"
photon polarization
horizontal
vertical
number of photons
vacuum
single photon
arrival time
early
late
phase quadrature
amplitude squeezing
phase squeezing
spin
-
+
number of electrons
absence
single electron
nucleus
spin (NMR)
-
+
optical lattice
atomic spin
-
+
Josephson junction
charge (charge qubit)
0
2e (one pair)
current direction (flux qubit)
clockwise current
counter clockwise current
energy (phase qubit)
base state
excited state
quantum dot pair
charge localisation
electron on left
electron on right
quantum dot
dot spin
-
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Fig. 3. Idea of a flux qubit
The third concept is depicted in Fig. 4. Qubit is tuned by current source and driven by high frequency field pulses. Its state is characterised by the energy of Cooper pairs. Phase qubit may be measured directly by a voltage sensor.
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Table 3. Short history of “qubit computers” [3] 1998 2 qubits (NMR), Oxford University 1998
3 qubits (NMR), Grover’s algorithm run
2000
5 qubits (NMR), Technische Universität München
2000
7 qubits (NMR), Los Alamos National Lab.
2001
Shor’s algorithm run
2005
8 qubits (qubyte), Österreichische Akademie der Wissenschaften
2007
28 qubits (???), D-Wave Systems, Inc.
Superconducting qubits There are three main concepts of qubits utilising Josephson junctions [4]. The idea of a qubit tuned by an electric potential of the superconducting Cooper pair box is presented in Fig 2. The state of qubit may be measured as the number of Cooper pairs crossing the junction (e.g. by means of Bloch transistor). reservoir
r
Cooper pair box gate
U
Φ
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Solid state technologies are the most promising and can overcome scaling problem. However, nuclear magnetic resonance (NMR) is the most advanced as yet (Table 3).
JJ
Another idea is shown in Fig. 3. The qubit is tuned by microwave frequency pulses of the magnetic flux and its state is represented as the direction of the current in this RF-SQUID loop. Measurement can be achieved by means of other SQUID, coupled LC tank circuit or Andreev interferometer.
Fig. 4. Idea of a phase qubit
There is also fourth idea: charge–flux qubit. It's characterised by another ratio of the charge energy to the coupling energy of a junction and it binds some aspects of the designs mentioned before. Conclusion Quantum computer may exponentially speed up the solution of some problems. Superconducting qubits are very promising: they are solid-state, they can be coupled in qubit registers, they can be manufactured as integrated circuits. The biggest challenge is to minimize decoherence of quantum information and overcome scaling problems. Still, everyone must remember words of Harold Weinstock: "Never use a SQUID when a simpler, cheaper device will do." REFERENCES [1] Hyperphysics Portal, http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html [2] E . R i e f f e l , W . P o l a k , An Introduction to Quantum Computing for Non-Physicists, arXiv:quant-ph/9809016, 19 Jan 2000 [3] L . J a c a k , Komputer kwantowy: nowe wyzwanie dla nanotechnologii, Postępy fizyki, tom 53D, 2002 [4] M . H . D e v o r e t , A . W a l l r a f f , J . M . M a r t i n i s , Superconducting Qubits: A Short Review, arXiv:condmat/0411174, 7 Nov 2004
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Author: dr inż. Leszek Jaroszyński, Politechnika Lubelska, Instytut Podstaw Elektrotechniki i Elektrotechnologii, ul. Nadbystrzycka 38a, 20-618 Lublin, E-mail:
[email protected].
Fig. 2. Charge qubit and its symbolic diagram
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