of |ψi in the basis {|ψ1i, ...,|ψNi} can be calculated by hψi|ψi. The entry ij of the matrix representationof observable O can be calculated byhψi|O|ψji. The unitary dynamics can be expressed by a unitary matrix applied to the quantum state vector.
Figure 3.1: Quantum circuit of IBM’s 5-qubit Quantum Experience’ online interface [69]. The lines each represent one qubit. The squares represent simple single qubit gates such as the X, Hadamard, phase shift andZ gate. The circles denote a controlled 2 qubit operation. The pink boxes at the end indicate which qubits are measured.
notions of quantum information claim that the power of quantum computers stems from the fact that qubits can be in a linear combination of 0 and 1. But also a classical random bit (i.e. a classical coin toss) have this property to a certain extent. This is why sampling algorithms are often the most suitable competitors to quantum routines. Two major differences stemming from the mathematical formalism of quantum theory are:
1. The coefficients of the linear combination in the quantum case can be complex, which means they can be negative. This allows for interference effects when the state of a qubit is evolved by unitary evolutions
2. Observables in quantum theory can be non-commutative
A quantum computer can be understood as a physical implementation ofn qubits with a precise control on the evolution of the state. A quantum algorithm is a targeted manipulation of the quantum system which can always be described by a unitary map from an initial state of the overall quantum system |ψi to a desired final state |ψ0i, with which an input-output relation is computed. An important theorem in quantum information states that any unitary can be approximated by a sequence of elementary unitaries which only act on the Hilbert spaces of one or two qubits [70]. Based on this insight, quantum algorithms are widely formulated as circuits of elementary unitaries orquantum gates (see Figure 3.1). A universal quantum computer consequently only has to know how to perform a small set of operations on qubits, just like classical computers are build on a limited number of logic gates. Runtime considerations usually count the number of quantum gates it takes to implement the entire quantum algorithm. Efficient quantum algorithms are based on unitaries whose decomposition grows at most polynomially with the input size of the problem.
As much as a classical bit is an abstract model of a binary system that can have many different physical realisations, a qubit can be used as a model of many different quantum systems. Some current candidates for implementations are superconducting qubits [71], photonic setups [72], ion traps [73] or topological properties of quasi-particles [74]. Each of them has advantages and disad- vantages, and it is not unlikely that future architectures use a mixture of these implementations.
Amplitude Index basis state shorthand a0 000 ˆ= 0 |000i |0i a1 001 ˆ= 1 |001i |1i a2 010 ˆ= 2 |010i |2i a3 011 ˆ= 3 |011i |3i a4 100 ˆ= 4 |100i |4i a5 101 ˆ= 5 |101i |5i a6 110 ˆ= 6 |110i |6i a7 111 ˆ= 7 |111i |7i
Table 3.1: The index of an amplitude from the 23-dimensional quantum state vector has a 3-bit binary representation that is exactly the basis state or event it is referring to.
3.2.2 Qubits and quantum gates
To recapitulate, a qubit is a quantum system with two possible measurement outcomes or possible events, and its state is an element of a two dimensional Hilbert spaceH2. The quantum state ofn qubits is an object inH2n=H2⊗. . .⊗ H2. Let{|ψii}be a basis of the joint Hilbert space, which consists of 2n=N basis states. A quantum state of ann-qubit system can then be expressed as
|ψi=
N−1
X
i=0
ai|ψii, with ai∈C,
N−1
X
i=0
|ai|2= 1.
In vector representation, the Hilbert space can be chosen as theC2n =C2⊗. . .⊗C2 with basis vectors{ai} and the quantum state becomes
a=
N−1
X
i=0
aiai, with a†a= 1.
Unless otherwise stated, I will use the standard basis when using the vector formulation, which corresponds to the computational basis in Dirac notation,
|0...0i ↔
1 ... 0
, . . . ,|1...1i ↔
0 ... 1
,
so that a quantum state is given by
a=
a0
... aN−1
in matrix notation and by
|ψi=a0|0i ⊗...⊗ |0i+. . .+aN−1|1i ⊗...⊗ |1i
=a0|0...0i+. . .+aN−1|1...1i
in Dirac notation. Running the indexifrom 0 toN−1 in the Dirac representation has the elegant
property that the amplitude at positionirefers to the event of measuring the qubits in then-bit binary representation of integeri(see Table 3.1). A generaln-bit quantum state in Dirac notation can therefore be compactly written as
|ψi=
2n−1
X
i=0
ai|ii. (3.1)
The corresponding density matrix is given by
ρpure=|ψihψ|=
N−1
X
i,j=0
a∗iaj|iihj|. (3.2)
For a general mixed state the coefficients do not factorise and we get
ρmixed =
N−1
X
i,j=0
aij|iihj|, aij ∈C. (3.3)
As a general rule, if there is an index between the Dirac bracket it refers to the corresponding basis state in computational basis.
As mentioned above, quantum gates are elementary unitary operators or building blocks from which larger algorithms can be built. The action of those operators onto quantum states is fully defined by their action on each eigenvector of an eigenbasis. It is practical to choose the computational basis as an eigenbasis. For example, using the basis{|0i,|1i}the single qubit Hadamard gate
H = 1
√2
1 1 1 −1
ˆ
= 1
√2(|0ih0|+|1ih0|+|0ih1| − |1ih1|) implements
H|0i → 1
√2(|0i+|1i) H|1i → 1
√2(|0i − |1i)
Another example is the single qubit gateσz
σz= 1 0
0 −1
ˆ
= |0ih0| − |1ih1|.
which writes a sign in front of qubits in the 1 state, while leaving qubits in the 0 state as they are.
The eigenstates ofσzare the computational basis states, σz|0i=|0i, σz|1i=−|1i.
The textbook by Michael Nielsen and Isaac Chuang is still an excellent introduction into qubits and gates [68].
E
x
Figure 3.2: Illustration of quantum annealing in an energy landscape over (here continuous) states or configurationsx. The ground state is the configuration of lowest energy (black dot). Quantum tunnelling allows the system state to transgress high and thin energy barriers (gray dot on the left), while in classical annealing techniques stochastic fluctuations have to be large enough to allow for jumps over peaks (gray dot on the right).
3.2.3 Quantum annealing and other computational models
Although the circuit model of qubits and gates is by far the most common formalism there are some other computational models. These have so far been shown to be equivalent up to a polynomial overhead, which means that efficient translations from one to the other are known.
Prominent in the quantum machine learning literature is a technique called quantum annealing, which can be understood as a heuristic to adiabatic quantum computing. Adiabatic quantum computing [75] is in a sense the analogue version of quantum computing [76] in which the solution of a computational problem is encoded in the ground state (i.e. lowest energy state) of a Hamiltonian which defines the dynamics of the system. Starting with a quantum system in the ground state of another Hamiltonian which is relatively simple to realise in a given experimental setup, and slowly adjusting the system so that it is governed by the desired Hamiltonian shall ensure that the ground state is found. Adjusting a Hamiltonian can be realised by changing field and interaction strengths between the physical objects that realise the qubits.
It turns out that for many problems, to keep the system in the ground state during the adjustment (‘annealing schedule’) requires a very slow evolution from one to the other Hamiltonian, and often a time exponential in the problem size, which shows once more that nature seems to set some universal bounds for computation. Quantum annealing may be seen as a heuristic or ‘shortcut’
to the adiabatic algorithm that works much like simulated annealing in computer science (see Boltzmann machine training in Section 2.3.2.4). The main difference between classical and quantum annealing is that thermal fluctuations are replaced by quantum fluctuations which enables the system totunnel through high and thin energy barriers (the probability of quantum tunnelling decreases exponentially with the barrier width, but is independent of its height). That makes quantum annealing especially fit for problems with a sharply ragged objective function (see Figure 3.2).
The great interest in quantum annealing is driven by the relatively mature hardware implementa- tion. Besides the hardware available in laboratories, a commercially available product, the D-Wave
U =eiHt |ψi= α0|00i+ α1|01i+ α2|10i+ α3|11i
amplitude encoding of unit-length complex vector (a0, a1, a2, a3)T basis encoding of binary string (10),e.g. to represent integer 2 Dynamic encoding of a Hermitian matrix
Figure 3.3: Illustration of the different encoding strategies for a quantum state of a 2-qubit system.
quantum annealer3, has been tested since a number of years by large research collaborations and gives insights into the practical challenges of quantum computing (which still seem comfortably dis- tant in the case of universal quantum computers). Current devices are limited to solving quadratic unconstrained binary optimisation (QUBO) problems,
(x1min,...,xN)
X
ij
wijxixj with xi, xj∈[0,1], wij∈R, (3.4)
where the ij couplings have to follow the connectivity of qubits in the hardware. Measuring the performance of quantum annealing compared to classical annealing schemes is a non-trivial problem, and although advantages of the quantum schemes have been demonstrated in the literature mentioned above, general statements about speed-ups are still controversial [77, 78].
Another famous quantum computational model isone-way or measurement-based quantum com- putation. The idea [79] is to prepare a highly entangled state called acluster stateand to perform a series of single-qubit measurements which conditionally depend on the output of former mea- surements. This computation is of course not unitary. The result can be either the state of the unmeasured qubits, or the outcome of a final measurement [80]. Many important quantum al- gorithms have been implemented using one-way computation [81, 82, 83, 84]. However, I do not know of any quantum machine learning approach based on this model, and it is an open question whether it offers a particularly suitable framework to approach supervised pattern recognition.