Quantum Machine Learning and Optimisation in Finance

1
The Principles of Quantum Mechanics

Quantum mechanics is a framework for the development of physical theories; it is not itself a physical theory [80]. Actual physical theories are built upon a foundation of quantum mechanics. This is why quantum mechanics plays such an important role in all natural sciences. Information theory is no exception and also derives inspiration from the ideas and methods of quantum mechanics.

Understanding quantum computing requires some familiarity with the basic principles of quantum mechanics. This book does not assume any prior knowledge of quantum mechanics and provides all the necessary definitions and explanations when needed. At the same time, the reader is encouraged to learn more about this fascinating subject at the level of mathematical formalism that she is comfortable with. Out of the extensive universe of textbooks on quantum mechanics that provide an introduction to this discipline, it is necessary to mention the classical book by Landau and Lifshitz [182] as well as the equally classical book on quantum computing by Nielsen and Chuang [223], which covers the most relevant aspects of quantum mechanics from the quantum computing perspective. For someone taking their first steps in quantum computing who would like to get the overall picture and some historical perspective, the excellent book by Bernhardt [32] provides both without the heavy usage of complex mathematical apparatus. Readers looking for a more formal modern take on the subject of quantum mechanics may find it in the book by Robinett [249]. The practical aspects of quantum computing are covered in great detail in the book by Sutor [278], and anyone looking for a python quantum computing programming textbook will find it in the work by Loredo [195].

1.1 Linear Algebra for Quantum Mechanics

Quantum computing and quantum mechanics rely on a specific notational formalism, due to Dirac, and are supported by classical linear algebra, in particular Hermitian structures of matrices and tensor products. We provide here a self-contained review of these tools to facilitate the understanding of the rest of the book. We start with basic linear algebra principles before introducing Dirac notations and the quantum counterparts of linear algebra tools. Sections 1.1.1 to 1.1.4 concentrate on standard definitions of finite-dimensional Hilbert spaces and matrices, while Sections 1.1.5 to 1.1.7 review the key details and properties of complex matrices (decompositions, Hermitian property, and rotations). Sections 1.1.9 to 1.1.11 introduce Dirac’s formalism and the essential aspects of quantum operators.

1.1.1 Basic definitions and notations

We let 𝔽 denote either the real field ℝ or the complex one ℂ. For a complex number z = x+iy ∈ℂ, with x,y ∈ℝ, we write the conjugate z^∗ := x−iy. We let ℳ_m,n(𝔽) denote the space of matrices of dimension m × n with entries in 𝔽 and ℳ_n(𝔽) whenever m = n. For A := (a_ij)_{1≤i≤m; 1≤j≤n} ∈ℳ_m,n(𝔽), A^∗ := (a_ij^∗)_{1≤i≤m; 1≤j≤n} is the complex conjugate. If A ∈ℳ_n(𝔽), we write A^⊤ for its transpose and A^† := (A^∗)^⊤ for its Hermitian conjugate. We finally denote I the identity matrix and write I_n whenever we wish to emphasise the dimension, and 0_m,n the null matrix in ℳ_m,n(𝔽). Recall that a matrix A ∈ℳ_n(𝔽) is invertible (or non-singular) if there exists B ∈ℳ_n(𝔽) such that AB = BA = I_n. Given two matrices A ∈ℳ_p,m(𝔽) and B ∈ℳ_q,n(𝔽), we define their tensor product as

⌊ ⌋ a11B ... a1mB || .. .. .. || A ⊗ B := ⌈ . . . ⌉ ∈ ℳpq,mn (𝔽). ap1B ... apmB

Since a vector is a particular case of a matrix, for u ∈𝔽^m and v ∈𝔽ⁿ, we can write

⌊ ⌋ |u1v1 | ⌊ ⌋ ⌊ ⌋ || ... || u1 v1 | | || .. || || ..|| ||u1vn || mn u ⊗ v = ⌈ . ⌉ ⊗ ⌈ .⌉ = ||u v || ∈ 𝔽 . um vn || 2.1 || |⌈ .. |⌉ umvn

1.1.2 Inner products

A vector space V over the field 𝔽 is a set endowed with

a commutative, associative addition operation,
an operation of multiplication by a scalar.

The addition and the multiplication by a scalar have the following properties (for scalars α,β ∈𝔽 and vectors u,v ∈V):

v + 0 = v;
v + (−v) = 0;
α(βv) = (αβ)v;
(α + β)v = αv + βv;
α(u + v) = αu + αv;
1 ⋅ v = v.

Armed with this, we can now define an inner product on V:

Definition 1. A map ⟨⋅,⋅⟩ : V ×V →𝔽 is called an inner product if, for u,v,w ∈V and α ∈𝔽,

(Positive definiteness) ⟨u,u⟩≥ 0 and ⟨u,u⟩ = 0 if and only if u = 0;
(Conjugate symmetry) ⟨u,v⟩ = ⟨v,u⟩^∗;
(Linear in the first argument) ⟨u + v,w⟩ = ⟨u,w⟩ + ⟨v,w⟩ and ⟨αu,v⟩ = α⟨u,v⟩;
(Antilinear in the second argument) ⟨u,v + w⟩ = ⟨u,v⟩ + ⟨u,w⟩ and ⟨u,αv⟩ = α^∗⟨u,v⟩.

The inner product is further called non-degenerate if ⟨u,v⟩ = 0 for all v ∈V ∖{0} implies u = 0.

For example, the following spaces carry a natural inner product:

The vector space ℂⁿ with the inner product ⟨u,v⟩ := u^†v = ∑ _i=1ⁿu_i^∗v_i;
The space of complex-valued continuous functions on [0,1] with ⟨f,g⟩ := ∫ ₀¹f(t)^∗g(t)dt;
If X,Y ∈ℳ_m,n(ℝ), then ⟨X,Y⟩ := Tr(X^⊤Y) = ∑ _i=1^m ∑ _j=1ⁿX_ijY_ij defines an inner product on the space of (real) matrices.

Projection matrices are particularly useful for geometric purposes:

Definition 2. A matrix P ∈ℳ_n(𝔽) is called a (orthogonal) projection if P² = P.

In particular, if W is a vector subspace of 𝔽ⁿ with some orthonormal basis (w₁,…,w_d), it is then easy to check that the map 𝒫_W : 𝔽ⁿ →𝔽ⁿ onto W satisfying

∑d 𝒫W (v) := ⟨v,wi⟩wi, for any v ∈ 𝔽n, i=1

defines an orthogonal projection.

1.1.3 From linear operators to matrices

Let V be a finite-dimensional vector space over 𝔽 and ⟨⋅,⋅⟩ a non-degenerate inner product on V. Given a linear operator 𝒜 : V →V, then, by the Riesz representation theorem [309, Section III-6], there exists a unique linear operator 𝒜^† : V →V, called the adjoint operator, such that

⟨𝒜u, v⟩ = ⟨u,𝒜 †v⟩, for all u,v ∈ V.

Indeed, for any v ∈V, the map u ∈V⟨𝒜u,v⟩ is a linear functional, hence an element of the dual space V^† (the space of bounded linear functionals on V), therefore for each v ∈V, there exists v′∈V such that ⟨𝒜u,v⟩ = ⟨u,v′⟩. It is then easy to show that the map vv′ is linear, proving that the adjoint operator is uniquely defined. In the particular case where 𝒜 = 𝒜^†, the operator 𝒜 is called Hermitian, a key requirement in quantum mechanics:

Definition 3. The operator 𝒜 is called Hermitian, or self-adjoint, if 𝒜 = 𝒜^†.

For a Hermitian operator 𝒜, we then have, for any u ∈V,

⟨𝒜u, u⟩ = ⟨u,𝒜 †u⟩ = ⟨u,𝒜u ⟩ = ⟨𝒜u,u⟩∗

by conjugate symmetry (Definition 1), and therefore ⟨𝒜u,u⟩ is real. Conversely, if ⟨𝒜u,u⟩ is real, then

∗ † ⟨𝒜u,u⟩ = ⟨𝒜u, u⟩ = ⟨u,𝒜u ⟩ = ⟨𝒜 u,u⟩.

Therefore, ⟨( ) ⟩ 𝒜 − 𝒜† u,u = 0; since this is true for all u ∈V, then 𝒜 = 𝒜^†.

The following property of operators shall be useful to ensure that systems driven by operators preserve distances, or norms:

Definition 4. The linear operator 𝒜 : V → V is called unitary if it is surjective and

⟨𝒜u, 𝒜v ⟩ = ⟨u,v ⟩, for all u, v ∈ V.

Recall that a linear operator between two finite-dimensional normed spaces is bounded, and therefore continuous. For any u ∈V, this implies that ∥𝒜u∥ = ∥u∥, so that a unitary operator 𝒜 preserves the norm. In that case, 𝒜 is an isometry, therefore injective. Being also surjective, it is bijective and therefore its inverse exists. For a unitary operator 𝒜 and any u,v ∈V, we have

† ⟨u,v⟩ = ⟨𝒜u, 𝒜v⟩ = ⟨u,𝒜 𝒜v ⟩

by definition of the adjoint, implying that

𝒜 †𝒜 = ℐ = 𝒜 𝒜 †,

where ℐ is the identity operator.

Example (Real Matrices): If V = ℝⁿ with inner product ⟨u,v⟩ := u^⊤v for u,v ∈ℝⁿ, the linear operator 𝒜 can now be viewed as a matrix A in ℳ_n(ℝ). Its adjoint is nothing other than the transpose A^⊤, and therefore A is self-adjoint if and only if it is symmetric. In this case, if A is unitary (or orthogonal), then it is invertible with A⁻¹ = A^⊤. Rotation matrices in ℝ², which will play an important role later when constructing quantum circuits, are the only unitary maps of ℝ² onto itself and are of the form

⌊ ⌋ cos(𝜃) δsin(𝜃) ⌈ ⌉, sin(𝜃) − δcos(𝜃)

for 𝜃 ∈ [0,2π) and δ ∈{−1,+1}.

Example (Complex Matrices): If V = ℂⁿ with inner product ⟨u,v⟩ := v^†u for u,v ∈ℂⁿ, the linear operator 𝒜 can now be viewed as a matrix in ℳ_n(ℂ). The adjoint of such a matrix is then the Hermitian conjugate A^† and A is called Hermitian if A = A^† and unitary if A^†A = I_n. We shall denote by 𝒰_n(ℂ) the set of unitary matrices in ℳ_n(ℂ). We will discuss Hermitian matrices over ℂ in more detail in Section 1.1.6.

1.1.4 Condition number

In order to manipulate matrices and measure them, we require matrix norms:

Definition 5. A matrix norm ∥⋅∥ : ℳ_m,n(𝔽) →ℝ is a function satisfying, for any α ∈𝔽 and A,B ∈ℳ_m,n(𝔽),

(positively valued) ∥A∥≥ 0;
(definite) ∥A∥ = 0 if and only if A = 0_m,n;
(absolutely homogeneous) ∥αA∥ = |α|∥A∥;
(triangle inequality) ∥A + B∥≤∥A∥ + ∥B∥.

The norm is further called sub-multiplicative if ∥AB∥≤∥A∥∥B∥.

The condition number of a matrix is an important tool to understand the stability of linear equations of the form Ax = b, for A ∈ℳ_n(𝔽), b ∈𝔽ⁿ. Assuming A to be non-singular, the true solution is clearly x_∗ := A⁻¹b. Suppose, however, that the vector b is only known up to some (not necessarily quantum) measurement error, and one observes instead b + Δ_b. The solution is then A⁻¹(b + Δ_b) = x_∗ + Δ_x, with Δ_x := A⁻¹Δ_b. In particular, we can write, for any (sub-multiplicative) matrix norm ∥⋅∥,

−1 ∥Δx-∥ = ∥A---Δb-∥≤ ∥A− 1∥ --∥b∥--∥Δb-∥ ≤ ∥A −1∥∥A∥ ∥Δb∥-. ∥x∥ ∥A −1b∥ ∥A− 1b∥ ∥b∥ ∥b ∥

From this inequality, we see that the quantity ∥A⁻¹∥∥A∥ bounds the relative error in the solution with respect to the relative error in the measurement of the input vector b. This leads to the following terminology:

Definition 6. Given a matrix A ∈ ℳ_n(𝔽) and a sub-multiplicative norm ∥⋅∥, we call

κ ∥⋅∥(A) := ∥A −1∥∥A ∥

the condition number (with respect to the norm ∥⋅∥) of the matrix A (and assign to it infinite value if A is singular).

Remark: The definition of the condition number above holds for any matrix norm ∥⋅∥, but admits a more explicit representation in the particular case of the spectral norm ∥⋅∥₂, defined as

∥A ∥2 := sup ∥Ax∥2-, x⁄=0 ∥x ∥2

where ∥x∥₂ := (∑ ) ni=1|xi|2 is the L ₂ norm for vectors. If the matrix A is not singular, then

|λ (A )| κ (A ) :=--max----, |λmin(A)|

where λ_max(A) and λ_min(A) denote the largest and smallest eigenvalues of A.

1.1.5 Matrix decompositions and spectral theorem

Having defined essential properties of (complex) matrices, we now introduce several essential tools that allow us to gain a better understanding of their properties.

The Singular Value Decomposition is a key tool to analyse the properties and behaviours of matrices. It is ubiquitous in applied statistics and machine learning and allows us to reduce the explanatory dimension of a large matrix into a small number of meaningful components.

Theorem 1 (Singular Value Decomposition). Let A ∈ℳ_m,n(𝔽) and p := min(m,n). There exist U ∈𝒰_m(𝔽), V ∈𝒰_n(𝔽) and σ₁ ≥ ⋅⋅⋅ ≥ σ_p ≥ 0 such that A = UΣV^†, where Σ ∈ℳ_m,n(𝔽) is diagonal with Σ_ii = σ_i for i = 1,…,p and Σ_ii = 0 for i > p.

The numbers {σ₁,…,σ_p} are called the singular values of A and are uniquely defined. The columns of U and V are the left-singular and right-singular vectors of A, in the sense that, if σ ∈{σ₁,…,σ_p}, then there exist a column u of U and a column v of V such that Av = σu and A^†u = σv. Recall that the rank of a matrix is defined as the dimension of the span of its columns. As a corollary of the Singular Value Decomposition theorem, the rank of a matrix is therefore equal to the number of non-zero singular values. The Singular Value Decomposition is general in the sense that it holds for any matrix. In the particular case of square matrices, the Schur decomposition and the Spectral Theorem provide refinements.

The Spectral Theorem is a cornerstone result in the theory of linear operators, and in particular for (finite-dimensional) matrices. Recall that an operator 𝒜 : V →V is called normal if it commutes with its adjoint, namely if 𝒜𝒜^† = 𝒜^†𝒜. Self-adjoint (or Hermitian) operators are clearly normal, yet the converse is not true in general. Recall further that an eigenvector of 𝒜 is a non-zero vector u ∈V such that 𝒜u = λu for some λ ∈ℂ, and we denote by σ(𝒜) the set of eigenvalues of 𝒜.

The following result, which is more general than the subsequent spectral theorem, allows us to decompose any arbitrary complex square matrix.

Theorem 2 (Schur Decomposition). For any A ∈ ℳ_n(ℂ) there exits a unitary matrix U ∈ 𝒰_n(ℂ) and an upper triangular matrix T such that A = UTU⁻¹.

Note that since U is unitary, then U⁻¹ = U^†. We call the matrix T the Schur transform of A and the identity in the theorem means that A and T are similar, so in particular, possess the same eigenvalues, all located on the diagonal of T. If A is a normal matrix, then so is T, and therefore T must be diagonal and we write T = D for clarity. In this case, we say that the matrix A is diagonalisable with A = UDU^†, where the diagonal entries of D are the eigenvalues of A and the column vectors of U are the orthonormal eigenvectors of A.

Theorem 3 (Spectral Theorem). The linear operator 𝒜 : V → V is normal if and only if there exists an orthonormal basis of V consisting of eigenvectors of A.

For each eigenvalue λ ∈ σ(𝒜), denote the corresponding eigenspace

𝒱λ := {u ∈ V : 𝒜u = λu} .

Since the vector space V is the orthogonal direct sum of the eigenspaces (indexed by the eigenvalues of 𝒜), we can then write the spectral decomposition

∑ 𝒜 = λ𝒫 λ, λ∈σ(𝒜)

where 𝒫_λ is the orthogonal projection operator onto 𝒱_λ. Note that such an operator is naturally self-adjoint [309, Theorem 2, Section III-1].

1.1.6 Hermitian matrices

We introduced above Hermitian matrices as the set of matrices A over the complex field ℂ such that A = A^†. As fundamental building blocks of quantum computing, we investigate their properties further. Clearly, a real matrix is Hermitian if and only if it is symmetric, in which case A^⊤ = A.

Proposition 1. The eigenvalues of a Hermitian matrix are real.

Proof. If Ax = λx for λ ∈ℂ and x ∈ℂⁿ, then

⟨Ax,x⟩	= x^†Ax = λx^†x = λ∥x∥²,
⟨x,Ax⟩	= (Ax)^†x = (λx)^†x = λ^∗x^†x = λ^∗∥x∥².

Since both are equal by the Hermitian property, then λ = λ^∗, proving the proposition. □

The Singular Value Decomposition (Theorem 1) takes a particular flavour in the case of Hermitian matrices:

Theorem 4. With the notations of Theorem 1, if A ∈ ℳ_n(ℂ) is Hermitian, then the matrices U and V are equal and the matrix Σ is diagonal with real entries.

Theorem 5. For a Hermitian matrix A ∈ ℳ_n(ℂ), the following are equivalent:

The eigenvalues are non-negative.
There exists a Hermitian matrix B ∈ℳ_n(ℂ) such that A = B².
There exists a matrix B ∈ℳ_n(ℂ) such that A = B^†B.
For every x ∈ℂⁿ, ⟨Ax,x⟩≥ 0.

Such a matrix is called positive semi-definite.

Proof. The Spectral Theorem shows that there exist a unitary matrix U ∈𝒰_n(ℂ) and a diagonal matrix Σ ∈ℳ_n(ℂ) such that A = UΣU^†, where the diagonal elements of Σ are the eigenvalues of A. Assuming (i), we can define B = U √ Σ- U^†∈ℳ_n(ℂ). Then clearly

† 2 ( √ -- †)( √ -- †) B = B and B = U ΣU U ΣU = A,

since U is unitary. The equality A = B^†B is also obvious. The latter implies that

† 2 n ⟨Ax, x⟩ = ⟨B Bx,x⟩ = ⟨Bx,Bx ⟩ = ∥Bx ∥ ≥ 0, for any x ∈ ℂ .

Finally, assume (iv) and let λ be an eigenvalue of A with eigenvector u. Then

2 ⟨Au, u⟩ = ⟨λu,u⟩ = λ⟨u,u⟩ = λ∥u∥ .

Since the latter is strictly positive, then clearly λ ≥ 0.

The following property lies at the core of Hamiltonian simulation of quantum systems:

Theorem 6. If A ∈ ℳ_n(ℂ) is Hermitian, then, for any t ∈ ℝ, e^itA is unitary; conversely, every unitary matrix has the form e^itA for some Hermitian matrix A.

Recall that for a matrix A ∈ℳ_n(ℂ), its exponential is given by

A ∑ Ak- e = k! . k≥0

In practice, though, given a Hermitian matrix A, finding the corresponding unitary matrix U is not easy. The Hamiltonian simulation problem is defined as follows.

Hamiltonian Problem: Given a Hermitian matrix A ∈ℳ_n(ℂ), a time t > 0, a tolerance level 𝜀 > 0, and some matrix norm ∥⋅∥, find a unitary matrix U such that ∥∥ itA ∥∥ U − e ≤ 𝜀.

1.1.7 Rotation matrices

Rotation matrices, and later their quantum gate equivalents will play a key role in building quantum circuits. Let us start with the following lemma:

Lemma 1. If a matrix A ∈ℳ_n(ℂ) is such that A² = I, then for any 𝜃 ∈ℝ,

ei𝜃A = cos(𝜃)I + isin(𝜃)A.

Proof. This follows directly from the series expansion

∑ xk ex = --, k≥0 k!

which has an infinite radius of convergence. □

Lemma 1 will prove essential for computational purposes. As simple examples, consider the following:

Exercise: Compute e^i𝜃A for A ∈{X,Y,Z} and 𝜃 ∈ℝ, where

⌊ ⌋ ⌊ ⌋ ⌊ ⌋ X = ⌈0 1⌉ , Y = ⌈0 − i⌉, Z = ⌈1 0 ⌉. 1 0 i 0 0 − 1

For any α ∈ [0,2π), consider now the map ℛ_α : ℝ² →ℝ² such that

( ) ( ) ℛ α r cos(𝜃),rsin(𝜃) := r cos(𝜃 + α),rsin(𝜃 + α ) ,

for any r ∈ ℝ and 𝜃 ∈ [0,2π),

which is basically a rotation of angle α and does not affect the norm of the input vector. To the map ℛ_α, we can associate the (rotation) matrix R_α such that ℛ_α(u) = R_αu for any u ∈ℝ². It is easy (exercise) to show the following:

Lemma 2. The matrix R_α has the form

⌊ ⌋ cos(α ) − sin(α) R α = ⌈ ⌉ . sin (α ) cos(α)

This representation is the general form of a rotation matrix in ℝ² (introduced in (1.1.3)).

Exercise: Write the matrices e^i𝜃A for A ∈{X,Y,Z} from the previous exercise as rotation matrices.

1.1.8 Polar coordinates

Recall that a point z = x + iy, with x,y ∈ℝ, lying on the unit circle can be written as z = e^i𝜃 with 𝜃 ∈ [0,2π). Indeed, simply let x = r cos(𝜃), y = r sin(𝜃) and add the constraint r = 1. Consider now a general vector u ∈ℂ² of the form

u = αe1 + βe2,

with α,β ∈ℂ such that |α|² + |β|² = 1. Here, (e₁,e₂) forms a basis of ℝ²:

⌊ ⌋ ⌊ ⌋ 1 0 e1 := ⌈ ⌉, e2 := ⌈ ⌉ . 0 1

In polar coordinates, we can then write

i𝜃α i𝜃β u = rαe e1 + rβe e2.

Note that arbitrary multiplication phases have no influence – a fact of key importance in quantum mechanics – because, for any γ ∈ℝ,

|eiγα|2 = (eiγα)∗eiγα = α ∗e−iγeiγα = α∗α = |α|2,

so that in fact, multiplying u by the global phase e^−i𝜃_α and letting 𝜃 := 𝜃_β −𝜃_α, we consider

i𝜃 u = rαe1 + rβe e2.

Write temporarily r_βe^i𝜃 = x + iy. Insisting on u being on the unit sphere further imposes ∥u∥² = 1, namely

1	= ∥u∥² = r_αe₁ + (x + iy)e₂^†r_αe₁ + (x + iy)e₂
	= r_αe₁^⊤ + (x − iy)e ₂^⊤r_αe₁ + (x + iy)e₂
	= r_α² + x² + y²,

since (e₁,e₂) is orthonormal. This is nothing more than the equation of the unit sphere. In polar coordinates, we can write

x = rsin(𝜃)cos(ϕ), y = rsin(𝜃)sin (ϕ ), rα = r cos(𝜃),

and clearly r = 1 since we are on the unit sphere. Therefore

u	= cos(𝜃)e₁ + sin(𝜃)cos(ϕ) + isin(𝜃)sin(ϕ)e₂
	= cos(𝜃)e₁ + sin(𝜃)e^iϕe ₂.

1.1.9 Dirac notations

Given a vector v ∈ℂⁿ, Dirac’s ket and bra notations read

⌊ ⌋ |v1| |v2| |v⟩ := || .|| and ⟨v| := [v∗1,v2∗,...,v∗n]. |⌈ ..|⌉ vn

With these notations, the operation ⟨u,v⟩ := ⟨u|v⟩ defines an inner product on ℂⁿ. The notation for the standard orthonormal basis in ℂⁿ is ( |i⟩ )_{i=0,…,n−1}, i.e.,

⌊ 1⌋ ⌊ 0⌋ ⌊ 0⌋ | | | | | | || 0|| || 1|| || 0|| |0⟩ := || ..||, |1⟩ := || ..|| , ... |n − 1⟩ := || ..||. ⌈ .⌉ ⌈ .⌉ ⌈ .⌉ 0 0 1

In coordinates, we can write, for any u,v ∈ℂⁿ,

|u⟩ = ∑ u |i⟩ and |v⟩ = ∑ v |i⟩, i i i i

and therefore,

∑ ⟨u,v⟩ = u ∗ivi. i

1.1.10 Quantum operators

In the language of Dirac’s notations, we can define the outer product |u⟩ ⟨v| (for u ∈U and v ∈V) as a linear operator from V to U, two vector spaces, as

( ) |u⟩ ⟨v| |w⟩ := ⟨v|w ⟩ |u⟩, for any w ∈ V.

In particular, |v⟩ ⟨v| is the projection on the one-dimensional space generated by |v ⟩ . Any linear operator can be expressed as a linear combination of outer products as

∑ 𝒜 = Aij |i⟩⟨j|, ij

where |i⟩ and |j⟩ are the standard basis vectors (1.1.9).

Similarly to the linear algebra setting above, we can define an eigenvector of a linear operator 𝒜 : V →V as a non-zero vector |v ⟩ such that

𝒜 |v⟩ = λ |v⟩

for some complex eigenvalue λ. Associated with any linear operator 𝒜, the adjoint operator 𝒜^† satisfies

⟨ ⟩ ⟨u|𝒜v ⟩ = 𝒜 †u|v .

Indeed, in the language of linear operators above, we have

⟨u,𝒜v ⟩ = ⟨𝒜v, u⟩∗ = ⟨v,𝒜 †u⟩∗ = ⟨𝒜 †u,v⟩,

by definition of the inner product (Definition 1).

1.1.11 Tensor product

Given two vector spaces U and V of dimensions m and n, the tensor product U ⊗V is a vector space of dimension mn. For u ∈U and v ∈V, we can form the vector |uv ⟩ := |u⟩ ⊗ |v⟩ ∈U ⊗V with the following properties:

= + , for any u′∈U;
= + , for any v′∈V;
α = = , for any α ∈ℂ.

Given the linear operators 𝒜 : U →U and ℬ : V →V, we can then define their tensor product as an operator 𝒜⊗ℬ on U ⊗V:

( ) 𝒜 ⊗ ℬ |uv ⟩ := |(𝒜u),(ℬv)⟩,

which can be represented in matrix form as A ⊗ B ∈ℳ_mn,mn(ℂ).

This Dirac formalism, fully anchored in (classical) linear algebra, now opens the gates to a proper dive into the foundations of quantum mechanics.

1.2 Postulates of Quantum Mechanics

Quantum mechanics states several mathematical postulates that a physical theory must satisfy. It turns out that the mathematics of quantum mechanics allows for more general computation: more general definition of the memory state in comparison with classical digital computing and a wider range of possible transformations of such memory states. A natural question arises: what is the reason for this superior mode of computation not being used until very recently? The answer is that although quantum mechanics was formulated almost a century ago (Paul Dirac’s seminal work "The Principles of Quantum Mechanics" [86] was published in 1930), the realisation of the rules of quantum mechanics in the computational protocol performed on classical digital computers requires an enormous amount of memory. Exponential gains in computing power are offset by exponential memory requirements.

In order to perform quantum computations efficiently, we need to use actual quantum mechanical systems, with their ability to encode information in their states. To illustrate this point, the state of a quantum system consisting of n quantum bits (qubits) can be described by specifying 2ⁿ probability amplitudes – this is a huge amount of information even for very small systems (n ∼ 100) and it would be impossible to store this information in classical memory. It took decades of technological progress before quantum processing units (QPUs) – devices that control quantum mechanical systems performing computations – became feasible.

Let us now proceed with the formulation of the mathematical postulates that lie at the foundation of quantum mechanics. These postulates specify a general framework for describing the behaviour of a physical system [80, 182, 249]:

How to describe the state of a closed system.
How to describe the evolution of a closed system.
How to describe the interactions of a system with external systems.
How to describe observables of a system.
How to describe the state of a composite system in terms of its component parts.

1.2.1 First postulate – Statics

Postulate 1. Associated to any physical system is a complex inner product space known as the state space of the system. The system is completely described at any given point in time by its state vector, which is a unit vector in its state space.

What is the importance of the first postulate from the quantum computing point of view? The answer is that quantum mechanics offers us a straightforward generalisation of the classical binary digit (bit). The classical bit is a two-state system with controlled transitions between them. As an example, we can use an electrical switch that can exist in one of the two discrete, stable states ("on" and "off"). Although electrical switches may seem an odd physical realisation of bits in the age of transistors, they illustrate an important point about computation in general: it is substrate independent. Exactly the same computational results can be obtained using electrical relays and CMOS transistors.

The quantum mechanical version of a bit, called a quantum binary digit (qubit), is a quantum mechanical two-state system. The first postulate of quantum mechanics tells us that the state of such a system can be represented mathematically by a unit vector in the two-dimensional complex vector space. This also means that such a system can exist in a superposition of basis states. Indeed, any vector |v⟩ in the two-dimensional complex vector space,

⌊ ⌋ α |v⟩ = ⌈ ⌉ , β

can be represented as a linear combination of the standard basis vectors:

⌊ ⌋ ⌊ ⌋ ⌊ ⌋ ⌈ α⌉ ⌈1⌉ ⌈0⌉ β = α 0 + β 1 , |v⟩ = α |0⟩ + β |1⟩.

Since the state vector is a unit vector, the coefficients α and β must satisfy

|α|2 + |β|2 = 1.

The coefficients α and β are probability amplitudes. Even though a qubit can exist in a superposition of basis states, once measured (see Postulate 3), its state collapses to one of the basis states: |α|² and |β|² give us the probability of finding the qubit, respectively, in states |0⟩ and |1⟩ after measurement.

One can draw an analogy with how the space of natural numbers, ℕ, can be extended to the space of real numbers, ℝ, and then to the space of complex numbers, ℂ. We have a much wider range of functions that can operate on and take values in ℝ and ℂ than in ℕ. Similarly, allowing the two-state system to exist in a superposition of states significantly extends the range of possible operators that can transform such states (i.e., perform computation).

For example, there is no Boolean function f that, when applied twice to a classical bit, would result in a NOT gate: f(f(0)) = 1 and f(f(1)) = 0. But there is such an operator in quantum computing. We can easily verify by direct calculations that the matrix

⌊ ⌋ 1 + i 1 − i M := 1-⌈ ⌉ , 2 1 − i 1 + i

applied twice to the basis vector |0⟩ would transform it to the basis vector |1⟩ , and applied twice to the basis vector |1⟩ would transform it to the basis vector |0⟩ . M is an example of a quantum logic gate – an operator that transforms the state of a qubit, thus implementing the computation.

Remark: The state space of a physical system can be infinite-dimensional. The quantum computing paradigm based on infinite-dimensional Hilbert spaces is called continuous-variable quantum computing, which is realised in, e.g., some photonic quantum computing systems. However, in the context of digital quantum computing, we will restrict our analysis to finite-dimensional state spaces.

The state of a qubit (the fundamental memory unit of quantum computing that generalises the concept of a classical bit) can be described mathematically as a unit vector in the two-dimensional complex vector space. Any physical system whose state space can be described by ℂ² can serve as an implementation of a qubit.

1.2.2 Second postulate – Dynamics

Postulate 2. The time evolution of a closed quantum system is described by the Schrödinger equation

iℏd-|ψ-(t)⟩ = ℋ |ψ(t)⟩, dt

where ℏ is Planck’s constant and ℋ is a time-independent Hermitian operator known as the Hamiltonian of the system.

The Hamiltonian of a quantum system is an operator corresponding to the total energy of that system, and its eigenvalues are the possible energy levels of the system. The knowledge of the Hamiltonian provides all the necessary information about system dynamics.

In the Schrödinger equation (1.2.2), the state |ψ(t1)⟩ of a closed quantum system at time t₁ is related to the state |ψ (t2)⟩ at time t₂ by a unitary operator 𝒰(t₁,t₂) that depends only on t₁ and t₂ via

|ψ(t2)⟩ = 𝒰 (t1,t2) |ψ(t1)⟩ ,

where 𝒰(t₁,t₂) is obtained from the Hamiltonian ℋ as

( ) iℋ-(t2-−-t1) 𝒰 (t1,t2) = exp − ℏ .

† ⟨u|𝒰 𝒰 |v⟩ = ⟨u|v⟩ .

A unitary operator is a complex generalisation of a rotation: unitary operators take an orthonormal basis to another orthonormal basis, and any operator with this property is unitary. In quantum mechanics, physical transformations such as rotations, translations and time evolution correspond to maps that take quantum states to other quantum states. These maps should be linear and preserve the inner product. This allows us to look at the unitary operators as the quantum logic gates implementing quantum computation protocols. Furthermore, unitary operators are invertible, a key property that ensures that quantum computing is reversible.

Quantum logic gates (quantum counterparts of the Boolean logic gates in classical computing) are unitary operators that transform quantum states, thus implementing the computation.

1.2.3 Third postulate – Measurement

Given a Hermitian operator 𝒜, the spectral theorem implies that the state |ψ⟩ of a system can be written as a superposition

∑N |ψ⟩ = αi |ψi⟩ , i=1

where the coefficients (α_i)_i=1,…,N are complex probability amplitudes, assumed to be normalised with ∑ _i=1^N|α_i|² = 1, and where ( |ψi⟩ )_i=1,…,N are eigenfunctions of 𝒜. The measurement postulate then reads as follows:

Postulate 3. If we measure the Hermitian operator 𝒜 in the state |ψ⟩ given in (1.2.3), the possible outcomes for the measurement are the eigenvalues (λ_i)_i=1,…,N of 𝒜, and the probability p_i to measure λ_i is given by p_i = |α_i|². After the outcome λ_i, the state of the system becomes

|ψ⟩ = |ψi⟩.

An immediate measurement in the same computational basis will deliver the same result without any uncertainty.

The quantum measurements are described by measurement operators (𝒫_i)_i=1,…,N, acting on the state space of the system with N possible outcomes. If the state of the system is |ψ⟩ before the measurement, then the probability of outcome i is

ℙ(i) = ⟨ψ |𝒫 †i𝒫i |ψ ⟩.

The measurement operators should also satisfy the completeness condition

∑N 𝒫 †i𝒫i = ℐ, i=1

where ℐ is the identity operator. This ensures that the sum of the probabilities of all outcomes adds up to 1.

These measurement operators are linear but not unitary. From the quantum computing perspective, we are interested in measurement operators that are projections (Definition 2) onto the computational basis, such as the standard orthonormal basis given by (1.1.9).

For example, the measurement operators for a single qubit can be defined as

⌊ ⌋ ⌊ ⌋ 1 0 0 0 𝒫0 := |0⟩⟨0| = ⌈ ⌉ and 𝒫1 := |1⟩⟨1| = ⌈ ⌉ . 0 0 0 1

𝒫₀	= ⟨0\|α + β = α⟨0\| + β⟨0\| = α,
𝒫₁	= ⟨1\|α + β = α⟨1\| + β⟨1\| = β.

The measurement postulate of quantum mechanics states that an immediate measurement in the same computational basis will deliver the same result without any uncertainty. The key words here are "the same computational basis". What would happen if the subsequent measurement is performed in another basis (the basis specified by another set of linearly independent unit vectors from the state space)? For example, assume that the qubit is in state

1 1 |ψ⟩ = √---|0⟩ + √---|1⟩. 2 2

Measuring |ψ⟩ in { |0⟩ , |1⟩ } computational basis will result in observing states |0⟩ and |1⟩ with equal probability 1∕2. Let us assume that we measured |0⟩ . The qubit state is now

′⟩ |ψ = 1 ⋅ |0⟩+ 0 ⋅ |1⟩.

If we repeat the measurement in the same { |0⟩ , |1⟩ } computational basis, we obtain state |0⟩ with probability 1 in accordance with the measurement postulate. However, had we measured state |ψ′⟩ in the Hadamard basis { |+ ⟩ , |− ⟩ }, given by

|+ ⟩ := √1-(|0⟩+ |1⟩) and |− ⟩ := 1√--(|0⟩ − |1⟩), 2 2

we would have equal probabilities of |+⟩ and |− ⟩ outcomes. Let us assume that we measured and the state of the qubit is now

′′⟩ |ψ = 0 ⋅ |+ ⟩+ 1 ⋅ |− ⟩ .

If we repeat the measurement of state |ψ ′′⟩ in the Hadamard basis { |+⟩ , |− ⟩ }, we obtain state with probability 1. But the state of the qubit is an equal superposition of states |0 ⟩ and |1⟩ from the { |0⟩ , |1⟩ } computational basis perspective and we have an equal chance of measuring either |0⟩ or |1⟩ in this basis.

Remark: The basis vectors |0⟩ and |1⟩ that form the standard computational basis can be transformed into the basis vectors |+⟩ and |− ⟩ that form the Hadamard basis by applying the following unitary operator (rotation), called the Hadamard gate:

⌊ ⌋ -1-⌈1 1 ⌉ H = √2-- 1 − 1 .

Chapters 6, 10 and 11 provide examples of applications of the Hadamard gate.

The measurement plays a crucial role in quantum computing. This is the process of collapsing a quantum state and reading out the classical information: measuring qubits encoding a quantum state will produce a classical bit string. The measurement process generates probabilistic outcomes. Therefore, we need to perform measurements on the same quantum state multiple times to generate a sufficiently large number of classical bit strings to produce reliable statistics.

The process of measurement describes the collapse of the quantum state due to contact with the environment. After measurement, the states of the qubits are known without any uncertainty. It is possible to extract at most 1 bit of information from a qubit. In order to extract more information about the probability distribution encoded in a given quantum state, it is necessary to perform measurement of the same state multiple times.

1.2.4 Fourth postulate – Observable

Postulate 4. For every measurable property of a physical system, there exists a corresponding Hermitian operator. The values of the physical observables correspond to the expectation values of Hermitian operators. The expectation value ⟨𝒜 ⟩ of the Hermitian operator 𝒜 in the normalised state |ψ ⟩ is given by

⟨𝒜 ⟩ := ⟨ψ|𝒜 |ψ⟩.

Let us consider the general case where the expectation value of a Hermitian operator 𝒜 is calculated in state |ψ ⟩ , which is not an eigenfunction of 𝒜. By the Spectral Theorem 3 (see also (1.2.3)), the state |ψ ⟩ of a system can be represented as the superposition

N ∑ |ψ⟩ = αi |ψi⟩ , i=1

where ( |ψi⟩ )_i=1,…,N are the eigenfunctions of 𝒜 and (α_i)_i=1,…,N the corresponding probability amplitudes.

Therefore, the expectation value of 𝒜 in state |ψ⟩ , given in (1.2.4), is calculated as

∑N ∑N N∑ ∑N ⟨𝒜⟩ = α∗iαj ⟨ψi|𝒜 |ψj⟩ = α ∗iαjλj ⟨ψi|ψj⟩, i=1j=1 i=1 j=1

where (λ_i)_i=1,…,N are the eigenvalues of 𝒜. The only terms that survive in the expression for ⟨𝒜 ⟩ are those with i = j due to the orthogonality of the eigenfunctions, so that

N N ∑ ∗ ∑ 2 ⟨𝒜 ⟩ = α iαiλi = |αi| λi. i=1 i=1

Therefore, the value of the observable is a weighted average of the eigenvalues of the corresponding Hermitian operator. The weights are the coefficients (|α_i|²)_i=1,…,N, which are the probabilities of measuring the corresponding eigenstate of 𝒜.

Hermitian operators play an exceptionally important role in quantum mechanics since their expectation values correspond to physical observables.

1.2.5 Fifth postulate – Composite System

Postulate 5. The state space of a composite physical system is the tensor product of the state spaces of the individual component physical systems.

If the first component physical system is in state |ψA ⟩ and the second component physical system is in state |ψB⟩ , then the state of the combined system, |ψ⟩ , is given by the tensor product

|ψ⟩ = |ψA⟩ ⊗ |ψB⟩ .

Not all states of a combined system can be separated into the tensor product of states of individual components. If the state of a system cannot be separated into component parts, we say that the component parts are entangled.

The entanglement of quantum systems is one of the major sources of computational power of quantum computing. It allows us to store exponentially more information in the correlations between the states of individual subsystems (in the limit – individual qubits) than directly in the states of individual subsystems.

To illustrate this point, we can look at the number of probability amplitudes needed to describe the state of an n-qubit system. An individual qubit can be found in one of the two possible states after measurement – one of the two basis states, |0⟩ or |1⟩ . This means that we need to specify two probability amplitudes to fully describe the state of the qubit before measurement. If all our qubits are independent and the state of the system can be represented as a tensor product of individual qubit states,

|ψ⟩ = |ψ1 ⟩⊗ |ψ2⟩⊗ ...⊗ |ψn ⟩,

then we need to specify 2n probability amplitudes (two for each individual quantum states) to describe the state |ψ ⟩ of the system. If, however, all individual qubits are entangled and the tensor product representation of the system state |ψ ⟩ does not exist, we need to specify 2ⁿ probability amplitudes – this is an effective measure of useful information that can be stored in the system.

The power of quantum computing is derived from the principles of superposition and entanglement. Entanglement allows us to store most of the information in correlations between the qubit states.

1.3 Pure and Mixed States

There are situations where the state of a quantum mechanical system cannot be described with the help of a state vector. Here, we look at such situations and provide a mathematical tool for describing them.

1.3.1 Density matrix

Let us start with the state of a combined two-component physical system given by (1.2.5). Let ( |i⟩ )_i=1,...,N and ( |j⟩ )_j=1,...,M denote, respectively, the standard orthonormal bases of the Hilbert spaces of systems A and B:

N M ∑ ∑ |ψA ⟩ = αi |i⟩, |ψB ⟩ = βj |j⟩, i=1 j=1

where (α_i)_i=1,...,N and (β_j)_j=1,...,M are some probability amplitudes. The states that allow the state vector representation (1.3.1) are called pure states. In this case, the state of the combined system is

∑N M∑ |ψ ⟩ = |ψA ⟩⊗ |ψB ⟩ = αiβj |i⟩⊗ |j⟩. i=1 j=1

However, in general, the state of the combined system would look like

∑N ∑M |ψ⟩ = γij |i⟩⊗ |j⟩, i=1 j=1

where γ_ij are probability amplitudes that may not necessarily be factorised as the product of probability amplitudes (α_i)_i=1,...,N and (β_j)_j=1,...,M. If γ_ij cannot be factorised as α_iβ_j, then the component systems A and B are entangled and their states cannot be represented by the state vectors (1.3.1). Such states of systems A and B are called mixed states.

The more general setup is that of an ensemble of states of the form {p_k, |ψk ⟩ }_k=1,…,N, where each |ψi⟩ is a quantum state whose wavefunction is known with certainty (although this does not necessarily provide full knowledge of the measurement statistics), and each p_k is the associated probability (not amplitude) in [0,1]. In order to define properly pure and mixed states, introduce the density operator as follows:

Definition 7. A density operator ρ is a positive semidefinite Hermitian operator with unit trace and takes the form

∑N ρ := pk |ψk⟩⟨ψk|, k=1

where ∑ _k=1^Np_k = 1 and ⟨ψ |ψ ⟩ k l equals 1 if k = l and zero otherwise.

Mathematically, such a density operator ρ corresponds to a density matrix (ρ_kl)_k,l=1,…,N such that

N ρ = ρ†, Tr(ρ) ≡ ∑ ρ = 1, ρ ≥ 0, for all k = 1,...,N. kk kk k=1

1.3.2 Pure state

A pure state is one that can be represented by a state vector

N∑ |ψ⟩ = αi |i⟩, i=1

where (α_i)_i=1,...,N are probability amplitudes in ℂ such that ∑ _i=1^N|α_i|² = 1. In the ensemble setup above, this means that there exists k^∗∈{1,…,N} such that p_k^∗ = 1 and hence |ψ⟩ = |ψk∗⟩ and therefore ρ = |ψ ⟩ ⟨ψ|. The density matrix also allows us to compute expectations of the form (1.2.4):

Lemma 3. Let ρ be the density matrix associated to the pure state (1.3.2) and let 𝒜 be an observable (Hermitian operator), then

⟨𝒜⟩ := ⟨ψ |𝒜 |ψ ⟩ = Tr (ρ𝒜 ).

Proof. The lemma follows from the immediate computation

⟨ψ\|𝒜	= ⟨ψ\|𝒜∑ _i=1^Nα _i
	= ∑ _i=1^Nα _i ⟨ψ\|𝒜
	= ∑ _i=1^N⟨ψ\|𝒜
	= ∑ _i=1^N ⟨i\|ρ𝒜 = Tr(ρ𝒜).

□

With the state |ψ ⟩ given by (1.3.2), we obtain

N∑ ∑N ⟨𝒜 ⟩ = αiα ∗j ⟨j|𝒜 |i⟩. i=1 j=1

At the same time we have

∑N ∑N ⟨𝒜 ⟩ = Tr(ρ𝒜) = ρij ⟨j|𝒜 |i⟩. i=1j=1

Comparison of (1.3.2) and (1.3.2) yields the following expression for the density matrix of a pure state:

∑N ∑N ρij = αiα∗, ρ = αiα∗ |i⟩⟨j| = |ψ ⟩⟨ψ |. j i=1 j=1 j

Example: An example of a pure state is the Hadamard state

⌊ ⌋ -1- -1- 1 |+ ⟩ = √2-(|0⟩+ |1⟩) = √2-⌈ ⌉, 1

with corresponding density matrix

⌊ ⌋ ρ = |+ ⟩⟨+ | = 1-⌈1 1⌉ . 2 1 1

1.3.3 Mixed state

A mixed state is one that cannot be represented by a single pure state vector, and is therefore represented as a statistical distribution of pure states in the form of an ensemble of quantum states {p_k, |ψk ⟩ }_k=1,…,N, where ∑ _k=1^Np_k = 1 and p_k ∈ [0,1] for each k. The density of a mixed state therefore reads

N ∑ ρ = pk |ψk ⟩⟨ψk|. k=1

Similarly to Lemma 3, we can write expectations of observables with respect to mixed states using the density matrix:

Lemma 4. Let ρ be the density matrix associated to the mixed state (1.3.3) and let 𝒜 be an observable (Hermitian operator), then

∑N Tr(ρ𝒜 ) = pk⟨ψk|𝒜 |ψk⟩ . k=1

Proof. The lemma follows from the immediate computation

Tr(ρ𝒜)	= ∑ _i=1^N ⟨i\|ρ𝒜
	= ∑ _i=1^N ⟨i\|𝒜
	= ∑ _k=1^Np _k
	= ∑ _k=1^Np _k ⟨ψ_k\|𝒜.

□

Let us see now how the density matrix formalism can help us describe the state of a combined system. Consider an entangled state of two systems, A and B, given by (1.3.1), and a Hermitian operator 𝒜 that only acts within the Hilbert space of system A. What would be the expectation value of 𝒜 in this state? Starting with (1.2.4), we obtain

N∑ ∑M ∑N M∑ ∗ ⟨𝒜 ⟩ = γijγ kl⟨k|𝒜 |i⟩⟨l|j⟩. i=1 j=1k=1 l=1

Since only terms with l = j survive in (1.3.3) due to the orthogonality of the basis states, we have

( ) ∑N ∑N ∑M ⟨𝒜 ⟩ = ( γijγ∗kj) ⟨k|𝒜 |i⟩. i=1 k=1 j=1

Thus, the density matrix that describes the mixed state of system A is

M ∑ ∗ ρik = γijγkj. j=1

Note that in the case where the probability amplitudes γ_ij can be factorised as the product of probability amplitudes (α_i)_i=1,...,N and (β_j)_j=1,...,M, we obtain

M∑ ∗ ∗ ∗∑M 2 ∗ ρik = αiβjαkβj = αiα k |βj| = αiαk, j=1 j=1

which describes a pure state.

A simple criterion to distinguish a pure state from a mixed state is the following:

Lemma 5. Let ρ be a density matrix. The inequality Tr(ρ²) ≤ 1 always holds and Tr(ρ²) = 1 if and only if ρ corresponds to a pure state.

Proof. Consider an ensemble of pure states {p_i, |ψi⟩ }_i=1,…,N, with density matrix given by (1.3.3). Therefore

Tr(ρ²)	= Tr
	= Tr
	= Tr = ∑ _i=1^Np _i²Tr⟨ψ_i\| = ∑ _i=1^Np _i² = ∑ _i=1^Np _i²,

which is smaller than 1 since the p_i are probabilities in [0,1] summing up to 1. Assume now that Tr(ρ²) equals one, then so does ∑ _i=1^Np_i². If p_i ∈ (0,1) for all i = 1,…,N, then

∑N 2 ∑N 1 = pi < pi = 1, i=1 i=1

which is a contradiction, and therefore there exists i^∗∈{1,…,N} such that p_i^∗ = 1, so that ρ = |ψi∗⟩ ⟨ψ_i^∗| is a pure state. Conversely, if ρ = |ψi⟩ ⟨ψ_i| for some i ∈{1,…,N} represents a pure state, then

Tr(ρ2) = Tr(|ψ ⟩⟨ψ | |ψ ⟩⟨ψ |) = Tr(|ψ ⟩⟨ψ |) = ⟨ψ |ψ ⟩ = 1. i i i i i i i i

□

Example: An example of a mixed state is a statistical ensemble of states |0⟩ and |1⟩ . If a physical system is prepared to be either in state |0⟩ or state |1⟩ with equal probability, it can be described by the mixed state

⌊ ⌋ 1- 1- 1-⌈1 0⌉ ρ = 2 |0⟩⟨0|+ 2 |1⟩⟨1| = 2 0 1 .

Note that this is different from the density matrix of the pure state

|ψ ⟩ = 1√--(|0⟩ + |1⟩), 2

which reads

⌊ ⌋ ρ = |ψ⟩⟨ψ | = 1(|0⟩+ |1⟩)(⟨0|+ ⟨1|) = 1(|0⟩⟨0|+|1⟩⟨0|+ |0⟩⟨1|+ |1⟩⟨1|) = 1⌈1 1⌉. ψ 2 2 2 1 1

Unlike pure quantum states, mixed quantum states cannot be described by a single state vector. However, the pure states and the mixed states can be described by the density matrix.

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Jason Saroni Dec 31, 2022

As an active quantum computing enthusiast who likes to participate in cutting edge quantum events, I had a chance to review the book and found the content super exciting with breadth and depth of interesting topics. It is comprehensive in the sense that it reviews the basic ingredients of linear algebra and quantum mechanics that are necessary for the current applications realized through quantum machine learning among other fields of inquiry. I am excited use the book as a reference that puts together valuable quantum computing information at a time when it is at its early stages of usefulness. The proofs are concise, important themes are discussed, and the range of applications is rewarding. The algorithms discussed can be implemented through one's favorite quantum hardware.

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Sadman Dec 22, 2022

I recently read the book "Quantum Machine Learning and Optimization in Finance" and I was thoroughly impressed. The authors did a fantastic job of explaining complex principles from Quantum Information Sciences, Computer Science, and Optimization Theory in a clear and concise manner.One of the things I appreciated most about the book was the way it seamlessly blended these different disciplines together to provide a comprehensive overview of quantum machine learning and optimization in finance. The authors clearly have a deep understanding of each subject and are able to explain the concepts in a way that is accessible to readers who may not have a background in all of these areas.The book also includes numerous examples and case studies to illustrate the concepts being discussed, which helped me better understand how these principles can be applied in real-world situations.Overall, I highly recommend "Quantum Machine Learning and Optimization in Finance" to anyone interested in learning more about the intersection of quantum information sciences, computer science, and optimization theory as applied to finance. It is a valuable resource for professionals working in the finance industry, as well as researchers and students studying these topics.

Jun Qi Dec 05, 2022

This book is a must-read one for the beginner to start the learning journey of quantum computing for machine learning, particularly in the application of finance. The first chapter concisely introduces the necessary foundations of quantum mechanics and important concepts of quantum computing. Then, the book reviews the adiabatic quantum computing protocol with optimization algorithms in finance. The gate model quantum computing is the core technique in this book, and the theoretical and application of quantum neural networks are comprehensively discussed in Part II.Overall, it is the best introductory book I have ever read for quantum machine learning and optimization algorithms in finance. Highly recommend!

Joydeep Dec 11, 2022

Even though the title of this book says "Quantum Machine Learning and Optimisation in Finance", which initially seems intimidating, but I must say, the book starts with very basic questions like "Why Quantum Computing" & "Why Quantum Machine Learning" and then built the knowledge base from ground up.There are multiple focus areas of this book starting from 'practical and real-world applications of Quantum Machine Learning (QML)' to 'hybrid quantum-classical computational protocols' to current 'major QML algorithms' which has shown signs of potential quantum advantage. The implementation of those knowledge has been presented mostly on the hardware-agnostic way and focuses on the details of 'fundamental characteristics of the algorithms'.This book takes finance domain to showcase how QML can be applied to NP-hard problems and it's practical use cases in finance like 'portfolio optimisation, credit card default prediction, credit approvals, and generation of synthetic market data' etc. This books seems to cater a vast user-base, starting from beginner to researchers to the professionals in the finance domain and presented the content in a very lucid way.From content point of view the coverage is vast which comprises of 'Linear Algebra & Matrix decompositions, Adiabatic Quantum Computing, QUBO problem, Quantum Boosting, Quantum Boltzmann Machine, Parameterised Quantum Circuits (PQCs), Quantum Neural Network (QNN), Quantum Circuit Born Machine(QCBM), Variational Quantum Eigensolver (VQE), Quantum Approximate Optimisation Algorithm etc'.From the implementation point of view, I think, due to it's hardware-agnostic way of explanation, the book does not cover any code samples either through qiskit/Q#/Cirq or any other quantum programing language, but I am sure during next editions, the authors will consider this as well.Overall the way authors covered the depth and breadth of the knowledge in this book is highly praiseworthy.

Siddhant Kochrekar Nov 26, 2022

The book came in pretty good condition, and I was waiting for this book to get published as I am taking a course in Applied Quantum Computing this semester. The reader is not expected to have any prior knowledge of Quantum mechanics, but the authors introduce the subject from the ground up.The authors have dedicated the first two chapters as a refresher in Quantum Computing. So the book gradually transits from mechanics to computing to applied Machine Learning and Finance. Although ML and Finance are the application domains in this book, you need to know the problem for which you are seeking QC solutions.In some parts, practical implications are lacking, but that is the shortcomings of this evolving field. But I recommend jumping in right now and catching the wave rather than waiting for the field and hardware to evolve fully. There are still a handful of executions shown on NISQ hardware. Power and practical extensions of analog quantum computers are succinctly displayed in this book. Certain blocks of text are assisted with pseudo-code, which makes it helpful while reading the material.They have compared QC with Machine Learning and Deep Learning algorithms in the form of clear analogies and explained in simple language suitable for all levels of readers. Quantum Boosting is my favorite part in this book as I have worked rigorously on XGBoost and other Boosting techniques in the Fintech industry. The book expands on ideas presented in research papers in the Quantum field and covers a few Finance case studies in Lending and Portfolio Management. Some advanced chapters have also tied QC with classical probabilistic approaches.On a high level, this book talks about all the topics from the angle of feasibility, error management, and resolution techniques. In the coming months, I would not be surprised if this text is used as supplementary reading material in applied cryptography courses in universities. I highly recommend this book to people studying Quantitative Finance.