Quantum computing was introduced in the s as a novel paradigm of computation, whereby information is encoded within a quantum system, as opposed to a system governed by the laws of classical physics. Wider interest in quantum computation has been motivated by Shor’s quantum algorithm for integer factorisation [shor1999polynomial], which provides an exponential speed-up over the best known classical algorithm for the same task. If implemented at scale, this would have severe security consequences for the ubiquitous RSA cryptographic protocol. Since then, a number of quantum algorithms demonstrating advantage over classical methods have been developed for a substantial variety of tasks; for a detailed survey, the reader is directed to [cleve1998quantum, montanaro2015quantum].
Consistent advances on both theoretical and experimental research fronts have meant that the reality of a quantum computer has been edging ever closer. Quantum systems are extremely sensitive to noise, with a primary challenge being the development of error correction in order to achieve fault tolerance [Campbell2017]. Nonetheless, the current advances observed over recent years in the quest to build a scalable universal quantum computer from both academic and industrial groups raise the question of applications of such a device.
Although powerful quantum algorithms have been devised, their application is restricted to a few use cases. Indeed, the design of a quantum algorithm directly relies on exploiting the laws and features of quantum mechanics in order to achieve a speed-up. More precisely, in quantum computing, a quantum state is first prepared, to which quantum operations are applied before final measurements are performed, thus yielding classical data. As discussed in [aaronson2015read], this model raises a number of challenges. In particular, careful consideration of how classical data can be input and obtained as output is crucial to maintaining the theoretical advantage afforded by quantum algorithms.
The question of solving a system of linear equations can be found at the heart of many problems with a wide scope of applications. An important result in recent years has been the Quantum Linear System algorithm (QLSA) [harrow2009quantum], also called Harrow-Hassidim-Lloyd (HHL) algorithm, which considers the quantum version of this problem. In particular, the HHL algorithm run on the quantum problem (that is, with quantum states as input and output) offers an exponential speed-up over the best known classical algorithm run on the classical problem. In the following, we present a complete review of the HHL algorithm and subsequent improvements for sampling from the solutions to linear systems. We have aimed to be as complete as possible, including relevant background where necessary. We assume knowledge of elementary linear algebra and some experience with analysis of classical algorithms.
1.2 Quantum linear systems algorithms
Solving a linear system is the task of taking a given matrix
and vectorand returning a vector satisfying . As a preview of what is coming up ahead, Table 1 shows the runtime of the best classical algorithm for solving linear systems, conjugate gradient (CG), compared with the quantum algorithms we shall introduce throughout these notes.
We note that CG solves a linear system completely, i.e. it returns the solution vector . The quantum algorithms allow one to sample from the solution efficiently, providing one has an efficient preparation method for the input state, i.e. a mapping from the vector to a quantum state .
|QLSP||Childs et. al. [childs2015quantum]|
1.3 Quantum computing
In this section, we introduce gate-model quantum computation, the computational model which will be used throughout. For a complete introduction to quantum computing we refer the reader to Nielsen and Chuang [nielsen2002quantum].
In classical computing, the input is a classical bit string which, through the application of a circuit, is transformed to an output bit string. This is achieved via the application of a finite number of classical gates picked from a universal gate set such as NAND. This framework is known as the classical circuit model of computation. In the case of quantum computing, there exist different yet equivalent frameworks in which the computation can be described: the quantum circuit model [nielsen2002quantum], measurement-based quantum computing (MBQC) [raussendorf2003measurement] and adiabatic quantum computing [farhi2000quantum]. In the following, we shall concentrate on the quantum circuit model, which is the closest quantum analogue of the classical circuit model as well as the model in which quantum algorithms are generally presented.
In classical computing the fundamental unit of information is the bit, which is either or
, whereas in quantum computing, it is the qubit,, such that and . This is represented by a two-dimensional column vector belonging to a complex Hilbert space . We shall denote by the conjugate-transpose of . The states and are basis vectors corresponding to a bit value of ‘0’ and ‘1’ respectively, and we can write and
. Thus, we have that a qubit is a normalised complex superposition over these basis vectors. Multiple qubits are combined using the tensor product, that is, for two qubits, their joint state is given by . Thus, an -qubit quantum state can be expressed as
where , and . Note that complex coefficients are required to describe a quantum state, a number growing exponentially with the system’s size. We call the basis the computational basis, as each basis vector is described by a string of bits.
There are two types of operations we can perform on a quantum state: unitary operators and measurements. A unitary operator has the property that , i.e. its inverse is given by the hermitian conjugate. Furthermore, this implies that the operator is norm-preserving, that is, unitary operators map quantum states to quantum states. A unitary operator on qubits can be expressed as matrix of dimension . Moreover, we have that unitary operators are closed under composition. A measurement is described by a collection of (not necessarily unitary) operators , where the index indicates a given measurement outcome. The operators act on the state’s Hilbert space and satisfy the completeness equation, . For a quantum state
, the probability of measuring outcomeis given by
and the resulting quantum state is then
The completeness equation encodes the fact that measurement probabilities over all outcomes sum to unity. A computational basis measurement, for consists of operators , the projectors onto the computational basis states.
In the circuit model of quantum computation, we are given an input , which is a classical bit string. The first step is to prepare an -qubit qubit quantum input state , where poly. A unitary operator is then applied to , and finally the output state is (customarily) measured in the computational basis – without loss of generality. The measurement outcome corresponds to a classical bit string , which is obtained with probability and which we refer to as the output of the computation.
In practice, a quantum computer will be built using a finite set of quantum gates which act on a finite number of qubits. Typically, we consider gates acting on either one or two qubits, leaving the others invariant. A set of quantum gates is said to be universal if any unitary operator can be approximated ‘well-enough’ using only gates from this set. More precisely, a set of gates is universal if any unitary operator can be decomposed into the sequence , such that , for any , where the . There are many such universal gate sets, such as for instance the Toffoli gate (which acts on three bit/qubits) and the Hadamard gate, or single-qubit rotations with a CNOT. Thus, any arbitrary unitary operator can be implemented given a universal set of gates.
This thus tells us that any arbitrary unitary operator can be approximated by the sequence to accuracy . But, how many gates are required to achieve a good accuracy? The Solovay-Kitaev theorem (see [nielsen2002quantum, Appendix 3]) states that , and thus exponential accuracy can be achieved using only a polynomial number of gates.
Finally, we discuss an important tool used in quantum computation, the oracle. Here, we are given a boolean function . The function is said to be queried via an oracle , if given the input (where and ), we can prepare the output , where denotes addition modulo . That is, the mapping
can be implemented by a unitary circuit , which takes the form
The effect of the oracle needs to be determined on all basis states, and the definition will always be given in terms of a state .
1.4 Quantum algorithms and machine learning
Quantum algorithms, in some cases, have the capacity to achieve significant speed-ups compared to classical algorithms. Most notably, classically, the prime factorization of an -bit integer using the general number field sieve is , where , and thus takes time exponential in the number of bits. In contrast, Shor’s factorization algorithm [shor1999polynomial] achieves an astonishing exponential speed-up with a polynomial runtime of . Another impressive result is the quadratic speed-up obtained by Grover’s algorithm for unstructured database search [grover1996fast, grover1997quantum], which we discuss in detail in section 2.7. These are just some examples of the many quantum algorithms which have been devised over the past decades [cleve1998quantum, montanaro2015quantum].
Machine learning [abu2012learning]
has had and continues to have a significant impact for artificial intelligence and more generally for the advancement of technology. Naturally, this raises the question of whether quantum computing could enhance and improve current results and techniques. The HHL algorithm[harrow2009quantum] considers the problem of solving a system of linear equations, which on a classical computer takes time polynomial in the system size . At its heart, the problem reduces to matrix inversion, and the HHL algorithm offers an exponential speed-up for this task, with a certain number of important caveats. This in turn raised the question of whether quantum algorithms could accelerate machine learning tasks, which is referred to as quantum machine learning (QML) – see the following reviews [aimeur2006machine, ciliberto2017quantum].
An interesting example which illustrates how quantum computing might help with machine learning tasks is quantum recommendation systems [kerenidis2016quantum]. Here, the netflix problem [koren2009matrix, bell2007lessons] is considered, whereby we are given users and films, and the goal is to recommend a film to a user which they have not watched, that they would rate highly given their previous rating history. The users’ preferences can be represented by a matrix of dimension , where the entry corresponds to the user’s rating of the film. Of course, the elements of are not all known, and the question is to provide good recommendations to the user. Classical algorithms for this problem run in time polynomial with matrix dimension, that is . In contrast, there exists a quantum algorithm with runtime complexity scaling as , thus providing an exponential speed-up [kerenidis2016quantum].
Another important example is the classical perceptron model, where we are given
labeled data points which are linearly separable and the goal is to find the separating hyperplane. Classically, we have that the number of training updates scales as, where is the margin, i.e. the shortest distance between a training point and the separating hyperplane. In contrast, the quantum perceptron model [wiebe2016quantum] exploits Grover’s search algorithm (see section 2.7), in order to achieve a quadratic speed-up .
Another important classical model for supervised learning is support vector machines (SVM), which are used for data classification and regression. For a special case of SVMs, known as least-squares SVMs, the classical runtime is polynomial in the number of training examplesand their dimension , , where denotes the accuracy. In contrast, quantum SVM [rebentrost2014quantum] offer an exponential speed-up in the dimensions and input number with a runtime of .
Finally, we note that quantum algorithms have also been developed for unsupervised learning, that is, in the case of unlabeled data[aimeur2013quantum, wiebe2014quantum], which also present a speed-up over their classical counterparts.
All of the algorithms mentioned here use the HHL – or some other quantum linear systems algorithm – as a subroutine.
We have aimed for a mostly modular presentation, so that a reader familiar with a particular subroutine of the HHL algorithm, say, phase estimation, can skip the relevant section if they wish. The structure of the text goes as follows.
, the quantum Fourier transform2.2, Hamiltonian simulation 2.3, quantum phase estimation 2.5, phase kickback 2.6, amplitude amplification 2.7, the uncompute trick 2.8 and finally quantum RAM 2.9. Next, in section 3, we present a detailed discussion of the HHL algorithm. We first formally define the problem in 3.1, before then discussing the algorithm in detail in 3.2, along with an error analysis in 3.3. In section 3.5 we consider the computational complexity of the problem, and in 3.5 its optimality. Then, in 3.6, the algorithm is extended to the case of non-Hermitian matrices.
Next, in section 4, we introduce modern updates to the algorithm, namely: improvements in the dependency on the condition number in 4.1; improvements on the precision number in 4.2 and in section 4.3, further improvements based on quantum singular value estimation giving an algorithm for dense matrices. More specifically, we discuss Jordan’s lemma and its consequences in LABEL:subsec:intersection_subspaces
, before reviewing the singular value decomposition in4.3.1 and finally presenting the quantum singular value estimation 4.3.2 and its application to linear systems in 4.3.3. This section deviates from the pedagogical style of the previous sections, giving an overview of the important ideas behind the results as opposed to all of the gory details.
2 Quantum algorithms: fundamental components
Here we review some of the fundamental ideas routinely used in the design of quantum algorithms, which will subsequently be applied in the HHL algorithm.
2.1 Notation and conventions
Any integer between and , where may be expressed as an -bit string , i.e. . Furthermore, given an integer , it is easy to verify that .
The Hadamard matrix is defined as . Given a vector , the Euclidean norm of x is . For an matrix with elements , the operator norm is given by , i.e., it is the sum of the Euclidean norms of the column vectors of . Note that in both cases this is equivalent to -norm. Next, we present the QFT.
2.2 Quantum Fourier transform
The QFT is at the heart of many quantum algorithms. It is the quantum analogue of the discrete Fourier transform, see 2.2.1, and is presented in section 2.2.2. In section 2.2.3, we will see how the QFT may be implemented efficiently using a quantum computer.
2.2.1 The discrete Fourier transform
The Fourier transform is an important tool in classical physics and computer science, as it allows for a signal to be decomposed into its fundamental components, i.e. frequencies. The Fourier transform tells us what frequencies are present and to what degree.
In the discrete setting, we have that the DFT is a square invertible matrixof dimension , where and . It is easy to show that the columns of this matrix are orthogonal and have unit length, and thus the set of column vectors form an orthonormal basis which we refer to as the Fourier basis. If the DFT is applied to a vector using matrix multiplication, then the time complexity scales as . Crucially, by using a divide and conquer approach this can be improved to , which is referred to as the fast Fourier transform (FFT).
2.2.2 The quantum Fourier transform
In a similar way, the QFT is defined by mapping each computational basis state to a new quantum state . The set of orthonormal states form an orthonormal basis set called the Fourier basis. The quantum Fourier transform with respect to an orthonormal basis is defined as the linear operator with the following action on the basis vectors:
The inverse Fourier transform is then defined as
But, what does this mean in terms of individual qubits? First, we represent the integer in binary notation, , and thus the Fourier basis states can be expressed as: . Expanding this expression, we have
Expanding the summation gives
Finally, the operation corresponds to the decimal expansion of up to bits, and we can thus write
We initially applied the QFT to the computational basis state . From Eq. (10), we see that information pertaining to the input is disseminated throughout the relative phase on each individual qubit. Thus, given the final state , applying the inverse QFT would yield the input string . Equivalently, one could obtain by performing a measurement in the Fourier basis.
2.2.3 Implementation of the QFT
The goal is to obtain the quantum state after applying a quantum circuit to an all-zero input state. From Eq. (10) we see that the state corresponds to a state where each qubit is initialised in the state and subsequently acquires a relative phase of , where , and where we recall that .
We now consider the quantum circuit which can implement this state. First, it is easy to see that a state of the form corresponds to either the state or depending on the value of . This can be expressed as , and can thus be obtained by the application of a Hadamard gate to a qubit in the state . Next, the state of the second qubit is given by , which can be re-expressed as . Thus, this corresponds to first preparing the state , and then applying a controlled rotation to the qubit, where the control is the th qubit in the state . Thus, the state on the first two qubits can be obtained by preparing the state , applying the Hadamard gate to the th qubit, and then a controlled rotation with qubit as control, where we have that is given by:
and controlled by qubit . Finally, SWAP operations are performed throughout for the qubits to be in the correct order. This approach can be extended to the qubits, where the number of gates scales as , and thus the QFT can be efficiently implemented in a quantum circuit.
2.3 Hamiltonian simulation
Most quantum algorithms for machine learning, and in particular the HHL algorithm, leverage quantum Hamiltonian simulation as a subroutine. Here, we are given a Hamiltonian operator , which is a Hermitian matrix, and the goal is to determine a quantum circuit which implements the unitary operator , up to given error. The evolution of a quantum state under a unitary operator is given, for simplicity, by the time-independent Schrödinger equation:
the solution to which can be written as .
Depending on the input state and resources at hand, there exists many different techniques to achieve this [berry2009black, berry2015hamiltonian, low2017hamiltonian, low2017optimal, low2016hamiltonian]. We give a brief introduction to this large field of still ongoing research, and interested readers can find further details in the lecture notes of Childs [childs2017lecture][Chapter V] and the seminal work [childs2003exponential].
The challenge is due to the fact that the application of matrix exponentials are computationally expensive. For instance, naive methods require time for a matrix, which is restrictive even in the case of small size matrices.
In the quantum case, the dimension of the Hilbert space grows exponentially with the number of qubits, and thus any operator will be of exponential dimension. Applying such expensive matrix exponentials has been studied classically, and in particular the classical simulation of such time-evolutions is known to be hard for generic Hamiltonians . As a consequence, new more efficient methods need to be introduced. In particular, a quantum computer can be used to simulate the Hamiltonian operator, a task known as Hamiltonian simulation, which we wish to perform efficiently. More specifically, we can now define an efficient quantum simulation as follows:
(Hamiltonian Simulation) We say that a Hamiltonian that acts on qubits can be efficiently simulated if for any , there exists a quantum circuit consisting of gates such that . Since any quantum computation can be implemented by a sequence of Hamiltonian simulations, simulating Hamiltonians in general is -hard, where refers to the complexity class of decision problems efficiently solvable on a universal quantum computer [kitaev2002classical].
Note that the dependency on is important and it can be shown that at least time is required to
simulate for time , which is stated formally by the no fast-forwarding theorem [berry2007efficient].
There are, however, no nontrivial lower bounds on the error dependency .
The hope to simulate an arbitrary Hamiltonian efficiently is diminished, since it NP-hard to find an approximate decomposition into elementary single- and two-qubit gates for a generic unitary and hence also for the evolution operator [shende2006synthesis, iten2016quantum, knill1995approximation]. Even more so, the optimal circuit sythesis was even shown to be QMA-complete [janzing2003identity]. However, we can still simulate efficiently certain classes of Hamiltonians, i.e. Hamiltonians with a particular structure.
One such example is the case when only acts nontrivially on a constant number of qubits, as any unitary
evolution on a constant number of qubits can be approximated with error at most using
one- and two-qubit gates, on the basis of Solovay-Kitaev’s theorem.
The origin of the hardness of Hamiltonian simulation stems from the fact that we need to find a
decomposition of the unitary operator in terms of elementary gates, which in turn can be very hard for generic Hamiltonians.
If can be efficiently simulated, then so can for any [childs2017lecture].
In addition, since any computation is reversible, is also efficiently simulatable and this must hence also hold for .
Finally, we note that the definition of efficiently simulatable Hamiltonians further extends to unitary matrices, since every operator corresponds to a unitary operator, and furthermore every unitary operator can be written in the form for a Hermitian matrix . Hence, we can similarly speak of an efficiently simulatable unitary operator, which we will use in the following.
2.3.1 Trotter-Suzuki methods
For any efficiently simulatable unitary operator , we can always simulate the Hamiltonian in a transformed basis , since
which follows from the fact that if is unitary, then we have that , which can easily be proven by induction. Another simple but useful trick is given by the fact that, given efficient access to any diagonal element of a Hamiltonian , we can simulate the diagonal Hamiltonian using the following sequence of operations. Let indicate a computational step, such that we can denote in the following a sequence of maps to a state:
In words, we first load the entry into the second register, then apply a conditional-phase gate and then reverse the loading procedure to set the last qubit to zero again.
Since we can apply this to a superposition and using linearity, we can simulate any efficiently diagonalisable Hamiltonian.
More generally, any -local Hamiltonian, i.e. a sum of polynomially many terms in the number of qubits that each act on at most qubits,
can be simulated efficiently. Indeed, since each of the terms in the sun acts only on a constant number of qubits, it can be efficiently diagonalised and thus simulated.
In general, for any two Hamiltonian operators and that can be efficiently simulated,
the sum of both can also be efficiently simulated, as we will argue below, first for the commuting case and then for the non-commuting case.
This is trivial if the two Hamiltonians commute. Indeed, we now omit the coefficient and consider for simplicity the operator . By applying a Taylor expansion, followed by the Binomial theorem and the Cauchy product formula (for the product of two infinite series), we have
Note that this is only possible since for the Cauchy formula we can arrange the two terms accordingly and do not obtain commutator terms in it. However (recall the famous Baker-Campbell-Hausdorff formula, see e.g. [rossmann2002lie]), this is not so for the general case, i.e. when the operators don’t commute. Here we need to use the Lie-Product formula [rossmann2002lie]:
If we want to restrict the simulation to a certain error , it is sufficient to truncate the above product formula after a certain number of iterations , which we will call the number of steps:
which, as we will show, can be achieved by taking , where we require that for the evolution to be efficiently simulable. To see this, observe that, from the Taylor expansion,
We need to expand a product of the form , where the operators and are non-commuting. Thus, we have that
where and do not commute in general. Specifically we have and and the first order terms in have the form
for . Next, we consider terms of order greater than one, where we have powers of for . Let us note that for , we have for . Thus, these first order and greater terms can be absorbed in the notation. Furthermore, in the following, we do not explicitly write the exponentials of the form as these will not play a role for bounding the error in the norm due to their unitarity. So, we continue to bound
In order to have this error less than , the number of steps must be .
This is a naive and non-optimal scheme. It can be shown that one can use higher-order approximation schemes, such that
can be simulated for time in for any positive but arbitrarily small [berry2007efficient, berry2009black].
These so-called Trotter-Suzuki schemes can be generalized to an arbitrary sum of Hamiltonians which then leads to an approximation formula given by
The following definitions are useful:
(Sparsity) An matrix is said to be -sparse if it has at most entries per row.
(Sparse matrix) An matrix is said to be sparse if it has at most entries per row.
Note that the sparsity depends on the basis in which the matrix is given. However, given an arbitrary matrix, we do not a priori know the basis which diagonalises it (and hence gives us a diagonal matrix with sparsity ), and so we need to deal with a potentially dense matrix, i.e. a matrix which has entries per row.
(Row computability) The entries of a matrix are efficiently row computable if, given the indices , we can obtain the entries efficiently, i.e. in time, where is the sparsity as defined above.
2.3.2 Graph-colouring method
Crucially, the simulation techniques described above can allow us to efficiently simulate sparse Hamiltonians. Indeed, if for any index , we can efficiently determine all of the indices for which the term is nonzero, and furthermore efficiently obtain the values of the corresponding matrix elements, then we can simulate the Hamiltonian efficiently, as we will describe below.
This method of Hamiltonian simulation is based on ideas from graph theory, and we will now first briefly introduce a couple of key notions relevant in our discussion. For further information, we refer the reader to the existing literature [west2001introduction]. An undirected graph is specified by a set of vertices and a set of edges, i.e., unordered pairs of vertices. When two vertices form an edge they are said to be connected. A graph can be represented by its adjacency matrix , where if the vertices and are connected, and , otherwise. The degree of a vertex is given by the number of vertices it is connected to. The maximum degree of a graph refers the maximum degree taken over the set of vertices. The problem of edge colouring considers if, given colours, each edge can be assigned a specific colour with the requirement that no two edges sharing a common vertex should be assigned the same colour. Vizing’s theorem tells us that, for a graph with maximum degree , an edge colouring exists with at most . Finally, a bipartite graph is a graph, where the set of vertices can be separated into two disjoint subsets and such that and no two vertices belonging to the same subset are connected, i.e., for every , we have , for .
Previously, we saw that a Hamiltonian operator can be represented by a square matrix. Thus, a graph can be associated with any Hamiltonian by considering the adjacency matrix with a at every non-zero entry of the Hamiltonian, and a elsewhere, in the spirit of combinatorial matrix theory. For a matrix of dimension , this will thus correspond to a graph with vertices. Previously, we saw that sparse Hamiltonians have at most polylog entries per row, and thus , for entries in total. This will translate into a graph having a number of edges .
Childs [childs2003exponential] proposed an efficient implementation for the simulation of sparse Hamiltonians by using the Trotterization scheme presented above (c.f. section 2.3.1) and a local colouring algorithm of the graph associated with the -sparse Hamiltonian. The core idea is to colour the edges of the Hamiltonian . Then, the Hamiltonians corresponding to each subgraphs defined by a specific colour can be simulated, and finally the original Hamiltonian recovered via the Trotter-Suzuki method [childs2003exponential].
More precisely, the first step is to find an edge-colouring of the graph associated with the Hamiltonian . This will be achieved using colours, which in the case of a sparse Hamiltonian, will be at most polylog. Next, the graph can be decomposed by considering the subgraphs corresponding to a single colour. We thus obtain a decomposition of the original Hamiltonian in a sum of sparse Hamiltonians, containing at most polylog terms. It is easy to convince oneself that each of these terms consists of a direct sum of two-dimensional blocks. Indeed, each adjacency matrix corresponding to a subgraph will be symmetric, with at most one entry per row, meaning that the evolution on any one of these subgraphs takes place
in isolated two-dimensional subspaces. Thus, each Hamiltonian term can be easily diagonalised and simulated using the diagonal Hamiltonian simulation procedure as given in Eq. (13).
A crucial step in this procedure is the classical algorithm for determining the edge colouring efficiently. Vizing’s theorem guarantees the existence of an edge colouring using colours. But, the question remains as to how this can be efficiently achieved. Indeed, even though we are given the adjacency matrix representation of the entire graph, we will now restrict ourselves to accessing only local information i.e. each vertex has only access to information regarding it nearest-neighbours. Finding an optimal colouring is an NP-complete problem. However, there are polynomial time algorithms that construct optimal colourings of bipartite graphs, and colourings of non-bipartite simple graphs that use at most colours. It is important to note that the general problem of finding an optimal edge colouring is NP-hard and the fastest known algorithms for it take exponential time.
We thus now present a local edge-colouring scheme achieving a -colouring (where we recall that is the maximum degree of the graph, i.e. the sparsity) for the case of a bipartite graph. This, using a reduction [childs2017lecture], is sufficient for the simulation of an arbitrary Hamiltonian. Crucially, this scheme is efficient if the graph is sparse, i.e. if polylog. We note that better schemes exist [berry2007efficient, berry2009black, berry2015hamiltonian] and can allow for polynomial improvements of the simulation scheme in comparison to the one given here.
(Efficient bipartite graph colouring [linial1987distributive, linial1992locality]) Suppose we are given an undirected, bipartite graph with vertices and maximum degree (i.e. each vertex is connected to a maximum of other vertices - the so called neighbours - which is similar to sparsity ), and that we can efficiently compute the neighbours of any given vertex. Then there is an efficiently computable edge colouring of with at most colours.
The vertices of are ordered and numbered from through . For any vertex , let denote the index of vertex in the list of neighbours of , ordered in increasing number. For example, let have the neighbours with the list of neighbours neighbours of . Then, we have that index, and index. Then define the colour of the edge , where is from the left part of the bipartition and is from the right for all and which have an edge. The colouring of this edge
is then assigned to be the ordered pair colourindex index. Recall that an edge colouring assigns a colour to each edge so that no two adjacent edges share the same colour. These assigned colours in form of the index-pairs give a valid colouring since if and have the same colour, then indexindex, so and similarly, if and have the same colour, then index index, so . ∎
Using this lemma we can then perform Hamiltonian simulation in the following manner. First we ensure that the associated graph is bipartite by simulating the evolution according to the Hamiltonian , a block-anti-diagonal matrix
The graph associated to this will be bipartite, with the same sparsity as [childs2017lecture]. Observe that simulating this reduces to simulating since
Without loss of generality let us assume now that has only off diagonal entries. Indeed, any Hamiltonian can be decomposed as the sum of diagonal and off-diagonal terms , which can then be simulated the sum using the rule given in Eq. (21). We can then, for a specific vertex and a colour , compute the evolution by applying the following three steps:
First we compute the complete list of neighbours of (i.e. the neighbour list and the indices) and each of the colours associated to the edges connecting with its neighbours, using the above local algorithm for graph colouring from Lemma 1.
Let denote the vertex adjacent to via an edge with colour . We then, for a given , compute and retrieve the Hamiltonian matrix entry . We can then implement the following quantum state i.e. three qubit registers in which we load the elements in the first one, in the second and then load the matrix element into the last one. More specifically, we here prepare the quantum state and then using the state preparation oracle (e.g. qRAM, see section 2.9), we obtain which can be done efficiently, i.e. in time , which is the time required to access the data.
We then simulate the (-independent, i.e. only depending on the local entry of but not the general matrix) Hamiltonians , i.e. the at most polylog Hamiltonians we obtain from the graph colouring. Note that this is a sparse Hamiltonian which acts only on the neighbours as described in the colouring step. The simulation is efficient, since can be diagonalised in constant time, as it consists of a direct sum of two-dimensional blocks. Next, we apply a scheme, described below, whereby each complex matrix entry is decomposed into a real part and imaginary part and simulated separately. We can then simulate the diagonalised Hamiltonians such that we implement the following mapping
where is a diagonal element of the diagonalised Hamiltonian . This can also be done in superposition.
The Hamiltonian to be simulated has complex entries, and can thus be decomposed in real and imaginary parts. Let denote the vertex connected to via an edge of colour . The original Hamiltonian had complex entries, and we can express the entry associated to vertex as a sum of a real part and imaginary part . If we assume that these can be loaded independently i.e. can be loaded separately, then we can introduce the oracles which allow for the following mappings to be implemented:
and similarly the inverse operations, for which it holds that , , since we have bitwise adding modulo in the loading procedure.
In order to simulate the complex entries we need to implement a procedure which allows us to apply both parts individually and still end up in the same basis-element such that these sum up to the actual complex entry, i.e. that we can apply and separately to the same basis state. We use multiple steps to do so.
Given the above oracles, we can similarly simulate the following Hermitian operations (note that this is not a unitary operation):
where we apply the Hermitian operators to multiply with (and ) and the swap operation to the registers, which can be implemented efficiently since the swap can be done efficiently. The operator is described in detail in [childs2003exponential] We then can implement the operator
where the sum is about all colours . This acts then on as , since
which can be confirmed using the fact that and since we have modulo addition, i.e. .
2.4 Erroneous Hamiltonian simulation
One might wonder what would happen with errors in the Hamiltonian simulator.
For example, imagine that we simulate the target Hamiltonian with simulator such as a quantum computer, and this simulator introduces some random error terms. This will be an issue for as long as we
do not have fully error corrected quantum computers, or if we use methods like quantum density matrix exponentiation which can have
errors in the preparation of the state we want to exponentiate (see [lloyd2013quantum, kimmel2017hamiltonian]). For example, this is relevant for a method called sample-based Hamiltonian simulation [lloyd2013quantum, kimmel2017hamiltonian] where we perform the quantum simulation of a density matrix , i.e. trace- Hermitian matrix, which can have some errors.
For a more in depth introduction and analysis see for example cubitt2017universal. Errors in the Hamiltonian simulator have been investigated in depth and here, we only want to give the reader some tools to grasp how one could approach such a problem in this small section. For more elaborate work on this we refer the reader to [cubitt2017universal].
In the following we will use so called matrix Chernoff-type bounds. Let us recall some results from statistics:
Theorem 1 (Bernstein [bernstein1927theory]).
Let be random variables and let
be random variables and let, such that is the expectation value and , is bounded and are independent, identically distributed copies. Then, for all ,
This fundamental theorem in statistics is making use of the independence of the sampling process in order to obtain a concentration of the result in high probability. In order to provide bounds for Hamiltonian simulation, we will need to use matrix versions of these Chernoff-style results. We will thereby make certain assumptions about the matrix, such as for instance that it is bounded in norm, and that the matrix variance statistic - a quantity that is a generalization of the variance - has a certain value. We now state first the result and then prove it.
(Faulty Hamiltonian simulator) Hamiltonian simulation of a Hamiltonian operator with a faulty simulator that induces random error terms (random matrices) with expectation value with bounded norm for all and bounded matrix variance statistic in each term of the simulation can be simulated with an error less than using steps with probability of at least
To prove this we will need a theorem that was developed independently in the two papers [tropp2012user, oliveira2009concentration]
, which is a matrix extension of Bernstein’s inequality. Recall that the standard Bernstein inequality is a concentration bound for random variables with bounded norm, i.e. it tells us that a sum of random variables remains, with high probability, close to the expectation value. This can be extended to matrices which are drawn form a certain distribution and have a given upper bound to the norm. We call a random matrix independent and centered if each entry is independently drawn from the other entries and the expectation of the matrix is the zero matrix.
Theorem 2 (Matrix Bernstein).
Let be independent, centered random matrices with dimensionality , and assume that each one is uniformly bounded
We define the sum of these random matrices , with matrix variance statistic of the sum being
where . Then
Let us then recall the error in the Hamiltonian simulation scheme from above.