Superconducting circuits for quantum computation

Leszek JAROSZYŃSKI Lublin University of Technology, Institute of Electrical Engineering and Electrotechnologies

Superconducting circuits for quantum computation Abstract. It appears that "quantum computers" are coming from science-fiction to the reality. Quantum computation shows outstanding efficiency in some numerical problems. Length of qubit registers grows noticeably last years, so 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 przenikają właśnie z kart literatury fantastyczno-naukowej do rzeczywistości. Algorytmy kwantowe wykazują znakomitą wydajność w przypadku niektórych problemów. Długość rejestrów qubitowych rośnie każdego roku tak 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 z kropkami kwantowymi. (Obwody nadprzewodnikowe dla obliczeń kwantowych).

Keywords: qubit, quantum computation, 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. Another 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. In 1925, Werner Heisenberg, Max Born and Pascual Jordan proposed "Matrix mechanics" – first complete definition of quantum mechanics. That opened a new era in physics [1, 2]. Qubit For practical reasons we are used to do calculations using binary devices. A bit can hold binary portion of information: either zero or one and nothing else. A qubit has also two basis states – we call them frequently ket zero |0> and ket one |1>. However, unlike a bit, qubit can hold a linear superposition of those two states (1) (1)

|0> θ

x |1> Fig.1. A qubit representation on a Bloch sphere

Quantum registers and quantum entanglement Quantum register consists of at least two qubits

ψ φ = (α1 0 + β1 1 ) ⊗ (α 2 0 + β 2 1 ) =

(3)

= α1α 2 00 + α1 β 2 01 + β1α 2 10 + β1 β 2 11

where (4)

2

2

2

For instance, register can hold individual numbers

0 ⊗ 0 = 00 = 0 d 0 ⊗ 1 = 01 = 1d 1 ⊗ 0 = 10 = 2 d

where: α, β – complex probability amplitudes such as

1 ⊗ 1 = 11 = 3d

2

α + β =1.

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 probability |α|2 and state |1> with probability |β|2. Nothing is certain, it's just probable. Neglecting here astonishing and still controversial manyworlds interpretation of quantum phenomena proposed by Hugh Everett in 1957, single qubit doesn't seem so interesting – only qubit register shows full potential of this idea.

170

2

α1α 2 + α1β 2 + β1α 2 + β1β 2 = 1 .

ψ =α 0 +β 1 ,

2

y

φ

(5)

(2)

|ψ>

z

or all of them "simultaneously" (saying that way just for the sake of simplicity)

(0 + 1 )= 2 2 1 = ( 00 + 01 + 10 + 11 ) = . 2 1 = ( 0 d + 1d + 2 d + 3d ) 2 1

(6)

(0

+ 1 )⊗

1

PRZEGLĄD ELEKTROTECHNICZNY (Electrical Review), ISSN 0033-2097, R. 85 NR 5/2009

Therefore 270-bit register can hold just one very big number and 270-qubit system represents in parallel more states than the number of atoms in the Universe. Quantum parallelism is just that fact which decides of the power of quantum algorithms. Quantum entanglement is a phenomenon where the quantum states of two or more qubits are bound together in the way that the condition of quantum register can't be described in terms of the states of its qubits. In contrast to "separable state", it's called "entangled state". The most known example is

ψ 00 = α 00 + β 11 ≠ ψ 1 ⊗ ψ 2

(7)

on time complexity – in domain where quantum algorithms show their astonishing potential. Table 1 contains an assessment of execution times of algorithms representing different time complexity classes. It's clearly visible that some polynomial, exponential and factorial algorithms are beyond the capacity of today's binary computers. Table 1. Execution time of exemplary algorithms time complexity

linear O(n) 2

quadratic O(n )

which belongs to Bell states.

n

exponential O(2 )

Measurement 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. When a qubit is measured the result is 0 with probability |α|2 leaving the state |0> or 1 with probability |β|2 leaving the state |1>. So, the measurement of single qubit gives one bit of information about α and β. Matter complicates when comes to registers. When first qubit of the simplest qubit register is measured

α1α 2 00 + α1 β 2 01 + β1α 2 10 + β1β 2 11

(8)

it may return 0 with probability 2

α1α 2 + α1 β 2

(9)

2

.

Alternatively, it may return 1 with probability 2

β1α 2 + β 1 β 2

2

leaving the state (12)

20 μs

30 μs

40 μs

100 μs

400 μs

900 μs

1.6 ms

1 ms

1s

17.9 min

12.7 d

3

3.6 s

77·10 y

β1α 2 10 + β 1 β 2 11 2

β1α 2 + β 1β 2

2

18

8.4·10 y

34

2.6·10 y

Quantum algorithms In 1985 David Deutsch suggested that all computational devices would ultimately be physical systems governed by laws of quantum physics. He formulated a new concept of Universal Quantum Computer, which should replace well known Universal Turing Machine. He showed one of the first quantum algorithms named after his surname and he opened discussion on quantum algorithms, which may probably outperform any known classical realizations for some problems. In Table 2, a comparison of time complexity of chosen algorithms has been presented.

classical complexity n

2

n=40

d – day, y – year.

name

α1α 2 00 + α1β 2 01

n=30

10 μs

O( 2 2 )

Simon's HSP

α1α 2 + α1 β 2

(11)

factorial O(n!)

n=20

Table 2. Comparison of quantum and classical algorithms [3]

2

leaving the state (10)

data size n=10

.

Another constraints on quantum manipulation are prohibition of quantum information cloning or erasure, and also occurrence of such phenomena as quantum information teleportation and dense coding. Computational complexity Let's assume that a computer executes an elementary operation in a constant time. The time complexity of an algorithm can be described as the number of elementary operations necessary for a solution in relation to the input data size. It can describe the worst case – the number of steps that algorithm requires at most with an arbitrary input – or the average case. Commonly, time complexity is an asymptotic measure for infinite data set. There is also space complexity which measures amount of memory consumed by an algorithm. However, let's focus

1 3

Shor's factoring

quantum complexity

O( n )

O(2 n )

O( n 2 )

O( n )

O( n )

Grover's searching

Daniel Simon discovered in 1994 the efficient quantum algorithm for solving the oracle function problem (hidden subgroup problem). It was the first algebraic poser for which quantum computations may be distinctly faster than their classical counterparts. Demonstrated in 1994, Peter Shor's algorithm is the realization of the large number factoring in polynomial time. It's based on the Quantum Fourier Transform. Classical algorithms for mentioned problem take exponential time, that guarantees strength of today's cryptographic security systems, for instance. In 1996, Lov Grover proposed the optimal quantum algorithm of fast searching through unsorted data outrunning any known classical one. This is the first and hopefully not last quantum algorithm, which may have a great practical meaning – algorithmic searching is performed very often in many programs. One must remember that quantum algorithms won't solve all common tasks of binary computers. On other hand, no one should forget that a quantum computer may give an answer in acceptable time when a classical computer would spend amount of time equal to the age of our Universe [2].


171

Implementations of qubits Quantum computer will most likely consist of: a quantum arithmetic logic unit (ALU), a quantum memory and a classical microprocessor system controlling operation of the device [4]. Quantum ALU will contain quantum gates performing operations on qubits: - the Hadamard gate (denoted as "H", which calculates a Walsch-Hadamard transform – maps qubit sharp basis states to their superposition with equal probability amplitudes), - identity gate ("I", a quantum "no operation", NOP), - qubit NOT (Pauli's "X" gate, a quantum negation), - qubit phase flip gate (Pauli's "Z" gate), - qubit NOT and phase flip gate (Pauli's "Y" gate), - qubit rotation gate by π/4 ("S"), rotation by π/8 ("T"), - two-qubit controlled-NOT ("CNOT", which negates target qubit only when control qubit holds |1>). Table 3. Qubit implementations [3, 5, 6] physical object

information support

"0"

"1"

photon

photon polarization

horizontal

vertical

number of photons

vacuum

single photon

arrival time

early

late

coherent light

phase quadrature

amplitude squeezing

phase squeezing

electron

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 localization

electron on left

electron on right

quantum dot

dot spin

-

+

In spite of the catchy and exaggerated abstract presented at the beginning, the problem of the manufacture of usable quantum computer is still open. There are some tested concepts of building the smallest and the most important items – qubits. They are achieved by different approaches. The basic information is shown in Table 3. Solid state technologies are the most promising and can overcome scaling problem. However, nuclear magnetic resonance (NMR) is the most advanced judging by the number of reports (Table 4). Even though the theory of the quantum computation is well defined and non-controversial, one must agree that a key to the future of quantum computers lays in a fully controllable multi-qubit register.

172

Table 4. Short history of “qubit computers” [3, 5, 6] 1998

2 qubits (NMR), Oxford University

1998

Grover’s algorithm run, 2 or 3 qubits (NMR)

2000

5 qubits (NMR), Technische Universität München

2000

7 qubits (NMR), Los Alamos National Lab.

2001

Shor’s algorithm run, 7 qubits (NMR), 15 = 3 · 5

2005

8 qubits (qubyte), ion trap, Österreichische Akademie der Wissenschaften and Universität Innsbruck

2007

28 qubits (still controversial), Josephson junctions, D-Wave Systems, Inc.

Superconducting qubits Cooper pairs confined in a superconducting elements exhibit features of so called artificial atoms [7]. They have discrete energy levels and can be excited to generate coherent quantum oscillations between these states. Natural atoms exist just in their ordinary configuration and may be controlled using high frequency electromagnetic waves. The artificial systems may be lithographically designed to meet a given purpose and then can be driven by electric potentials, electric currents or microwave radiation. Josephson junction consists of two superconductors separated by a extremely thin insulating layer (~1 nm). Cooper pairs can experience quantum mechanical tunneling through insulator without breaking the pairs [1]. The main challenge is to limit fluctuation of the residual offset charge of the junction preserving nonlinearity of the device. That's the one of reasons why there are three main concepts of qubits utilizing superconductors – namely Josephson junctions [8]. The idea of a qubit tuned by an electric potential of the superconducting Cooper pair box is presented in Fig 2. When the offset charge induced by the gate potential is nearly the charge of a single electron, this system can be considered as a two-level quantum system. The state of qubit may be measured as the number of Cooper pairs crossing the junction e.g. by means of Bloch transistor. This circuit is characterized by the ratio of Josephson coupling energy to the electrostatic energy of order of magnitude -1. reservoir JJ

r

Cooper pair box gate

U

i g

Fig.2. Charge qubit (Cooper pair box) and its symbolic diagram

Another idea is shown in Fig. 3. The qubit is tuned by microwave frequency pulses of the magnetic flux which induce supercurrent in its loop. The state of this qubit is represented as the direction of the supercurrent: clockwise and counterclockwise. Approximate order of magnitude of ratio EJ/EC equals 1. Flux qubits are constructed as singleor multi-junction circuits. Measurement can be achieved by means of other SQUID, determination of the impedance of coupled LC tank circuit or by Andreev interferometer.


magnitude 4 is necessary for optimal quantum computation. The main reason of decoherence is an interaction with the environment, namely quantum 1/f noise which origin still eludes researchers. Construction of a "noise-proof" multijunction superconducting qubits may help to overcome mentioned limitation and increase dramatically decoherence time [9]. There is a second option – to design and to use algorithms enhanced by quantum error correction. This concept permits to reconstruct the exact quantum state of a qubit register [3].

M Φ

i

Fig.3. Idea of a flux qubit

The third concept is roughly depicted in Fig. 4. Qubit constructed of a large Josephson junction is biased by current source. Biasing current bends the characteristic curve of Josephson potential reducing the number of bound states in the potential well. System is driven by high frequency magnetic field pulses. Its state is characterized by the energy of Cooper pairs. Order of magnitude of EJ/EC ratio is around 6. Phase qubit may be measured directly by a voltage sensor.

Fig.4. Idea of a phase qubit

There is also fourth idea: charge–flux qubit (Fig. 5). It's characterized by the ratio of the coupling energy of a junction to the charge energy equal 1. It binds some aspects of the designs mentioned before.

M

i

Φ

U

Fig.5. Idea of a charge-flux qubit

Two coupled qubits form minimal qubit register necessary for some quantum gate operation and quantum computation at all. An easiest way to achieve an interaction of superconducting subcircuits is to use capacitive or inductive coupling [7, 8]. Relaxation of energy and variation of the phase in twolevel quantum system (dephasing, decoherence) are the most challenging aspects of superconducting qubit design. The second effect is more problematic since the ratio of decoherence time to single operation time of order of

Conclusion In a dozen of years the Moore's law will fail – the process of the IC production will approach quantum dimensions (~40 nm as yet). Microprocessor designers will have to deal with single electron charge or spin manipulation and therefore with quantum phenomena, one way or another. Quantum computers may exponentially speed up solutions of some problems, however a path to a working quantum computer seems to be quite long. The biggest challenge is to overcome scaling problems (long quantum registers) and minimize decoherence of quantum information. Superconducting qubits are fairly promising: - they are solid-state, - they can be coupled in qubit registers like any electrical circuits, - they can be manufactured as integrated devices, - they operate at cryogenic temperatures, that limits some sources of decoherence. 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] D e u t s c h D . , Lectures on quantum computation, http://www.quiprocone.org [3] R i e f f e l E . , P o l a k W . , An Introduction to quantum computing for non-physicists, arXiv:quant-ph/9809016, 19 Jan 2000 [4] O s k i n M . , C h o n g F . T . , C h u a n g I . L . , A practical architecture for reliable quantum computers," Computer, vol. 35, no. 1, Jan 2002 [5] J a c a k L . , Komputer kwantowy: nowe wyzwanie dla nanotechnologii, Postępy fizyki, tom 53D (2002) [6] M e h r i n g M . , Principles of quantum computing, The 21st Winter School on Electronic Properties of Novel Materials (IWEPNM 2007), 10-17 March 2007, Kirchberg, Austria [7] Y o u J . Q . , N o r i F . , Superconducting circuits and quantum information, Physics Today 58 (2005) 11 [8] D e v o r e t M . H . , W a l l r a f f A . , M a r t i n i s J . M . , Superconducting qubits: A Short Review, arXiv:condmat/0411174, 7 Nov 2004 [9] U s m a n o v R . A . , I o f f e L . B . , Theoretical investigation of a protected quantum bit in a small Josephson junction array with tetrahedral symmetry, Phys. Rev. B, 69 (2004) Issue 21 Author: dr inż. Leszek Jaroszyński, Lublin University of Technology, Institute of Electrical Engineering and Electrotechnologies, ul. Nadbystrzycka 38a, 20-618 Lublin, E-mail: [email protected].


173