Quantum Fibrations: quantum computation on an arbitrary topological space

Throughout this article, any Hilbert space we consider is separable and complex. All operators on any Hilbert space are assumed to be trace class and bounded. Let $B(\mathcal{H})$ be the set of all bounded operators on a Hilbert space $\mathcal{H}$. For $A\in B(\mathcal{H})$, we write its conjugate as $A^{*}$. Let $D(\mathcal{H})=\{\rho\in B(\mathcal{H}):\Tr\rho=1,\rho^{*}=\rho,\rho\geq 0\}$ be the set of all density operators (quantum states) acting on $\mathcal{H}$. Let $\rho$ be an endomorphism of $\mathbb{C}^{2^{n}}$ such that $\Tr\rho=1$, $\rho^{*}=\rho$ and $\rho\geq 0$. Such an operator $\rho$ is called a quantum state or density operator. A quantum state is called pure if $\Tr(\rho^{2})=1$; otherwise, it is called mixed. Let $\{E^{k}_{j}\}_{k=1,j=1}^{K,J_{k}}$ be a set of endomorphisms (Kraus operators) of $\mathbb{C}^{2^{n}}$ such that

(1.1)

$$\sum_{k=1}^{K}\sum_{j=1}^{J_{k}}(E^{k}_{j})^{*}E_{j}^{k}=I_{2^{n}}.$$

The measurement of the operator $\rho$ is defined using $\{E^{k}_{j}\}_{k=1,j=1}^{K,J_{k}}$ and the result $k$ can be obtained with probability

(1.2)

$$\Tr\left(\sum_{j=1}^{J_{k}}E^{k}_{j}\rho(E^{k}_{j})^{*}\right).$$

If the measurement result is $k$, the state $\rho$ is transformed into another state

(1.3)

$$\rho^{\prime}=\frac{\sum_{j=1}^{J_{k}}E^{k}_{j}\rho(E^{k}_{j})^{*}}{\Tr\left(\sum_{j=1}^{J_{k}}E^{k}_{j}\rho(E^{k}_{j})^{*}\right)}.$$

Quantum computation $\left(\rho_{in},\{U_{\lambda}\}_{\lambda\in\Lambda},\{E^{k}_{j}\}_{k=1,j=1}^{K,J_{k}}\right)$ is defined by an initial quantum state $\rho_{in}$, a family of unitary operators $\{U_{\lambda}\}_{\lambda\in\Lambda}$ and Kraus operators $\{E^{k}_{j}\}_{k=1,j=1}^{K,J_{k}}$. The initial state $\rho_{in}$ which is updated by sequentially applying unitary operators $\{U_{\lambda}\}_{\lambda\in\Lambda}$ as $\rho_{in}\mapsto U_{\lambda}\rho_{in}U^{*}_{\lambda}$. Quantum computation is called universal if any $2^{n}$-dimensional unitary operator can be approximated using several operators from $\{U_{\lambda}\}_{\lambda\in\Lambda}$.

The physical system that performs quantum computation is realized by graphically arranging $n$ artificial atoms, called qubits. Each qubit is an unit element of $\mathbb{C}^{2}$. Although theoretical methods for realizing universal quantum computation are well established, it is also well known that it is very difficult to simulate arbitrary physical processes by lattice computation. To explain this issue more precisely, let us illustrate the general procedure of quantum computation.

Let $(\Omega,\mathscr{F},P)$ be an arbitrary probability space where a given problem is defined. It can be solved by quantum computation in the following way:

• 1.

  Assign $\mathbb{C}^{2^{n}}$ and $\left(\rho_{in},\{U_{\lambda}\}_{\lambda\in\Lambda},\{E^{k}_{j}\}_{k=1,j=1}^{K,J_{k}}\right)$, where each $U_{\lambda}$ is generated by $n$-qubit Pauli operators.

• 2.

  For a given $\epsilon>0$, find an embedding map $f:\Omega\to\left(\rho_{in},\{U_{\lambda}\}_{\lambda\in\Lambda},\{E^{k}_{j}\}_{k=1,j=1}^{K,J_{k}}\right)$ such that for all $\omega\in\Omega$, there exists $i,\mu$ that satisfy $\left|P(\omega)-\Tr(\sum_{j=1}^{J_{i}}E^{i}_{j}U_{\mu}\rho_{in}U^{*}_{\mu}(E^{i}_{j})^{*})\right|<\epsilon$.

The technical difficulties in this process are as follows:

• (A)

  A sufficient number of qubits, which give $\mathbb{C}^{2^{n}}$, and operators $\left(\rho_{in},\{U_{\lambda}\}_{\lambda\in\Lambda},\{E^{k}_{j}\}_{k=1,j=1}^{K,J_{k}}\right)$ should be given to embed a problem and achieve a given precision $\epsilon$.

• (B)

  An embedding map $f$ should be found so the original distribution $P(\omega)$ can be approximated by $n$-qubit Pauli operators within a given precision $\epsilon$.

The former problem (A) is an experimental difficulty, while the latter (B) involves a theoretical difficulty. For example, to consider the Standard Model of elementary particles from a view point of lattice gauge theory, it is necessary to solve (circumvent) several ”No-go theorems”, such as the Nielsen-Ninomiya theorem, which states that right handed and left handed quarks and leptons appear in pairs, so chiral symmetry is not realized on a lattice. The construction of lattice fermions with exact chiral symmetry was circumvented by Neuberger’s proposal of overlap fermions, but the problem of huge computational cost for simulations appeared. Quantum computation is expected to help reduce such computational costs, but it is generally very difficult to equivalently replace a problem on a space having cardinality of the continuum $2^{\aleph_{0}}$ (or greater cardinality) with a problem on a graph/lattice having cardinarity of $\aleph_{0}$ or less. When we say that quantum computation is universal, we mean that it can produce arbitrary probability distributions or unitary operators with arbitrary precision (if we are allowed to use sufficiently long time and a large amount of memory space), but nothing is told about how this is possible.

One of the finest aspects of quantum computation is that, in principle, it can approximate arbitrary quantum many-body systems defined on any topological space in a well-defined manner. Quantum mechanics is generally described by operators acting in an infinite-dimensional complex Hilbert space, but in quantum computation they can be approximated by a finite number of operators acting in a finite-dimensional complex linear space. This gives an advantage when simulating quantum field theory and quantum gravity. However, embedding quantum field theory involves technical difficulties, as mentioned earlier. To solve these problems, it is natural to consider a model of quantum computation based on the quantum theory of fields. However, in [2], it is reported that topological quantum field theory (TQFT) cannot be used to define a model of computation stronger than BQP. The above state of affairs prompts the following question.

The most general framework that can address any quantum theory will be given as follows.

Let $\mathscr{F}$ and $X$ be topological spaces, and $(\mathscr{F},\pi,X)$ be a triple that satisfies the following conditions:

1. (1)

   $\pi:\mathscr{F}\to X$ is continuous.

2. (2)

   For each open cover $\{U_{\lambda}\}_{\lambda\in\Lambda}$ of $X$, $\pi^{-1}(U_{\lambda})$ is a set of quantum states for every $\lambda\in\Lambda$.

First of all, in the following way, one can check this framework is the most general one that can address any quantum theory. As we see below, continuity of $\pi:\mathscr{F}\to X$ is not a strong condition. Given a topological space $(X,\mathcal{O}(X))$, for any set $\mathscr{F}$ and any map $\pi:\mathscr{F}\to X$, we can define a topology into $\mathscr{F}$ so that $\pi$ is a continuous map. Let $\mathcal{O}_{\pi}(\mathscr{F})=\{U\subset\mathscr{F}:\exists V\in\mathcal{O}(X),~{}U=\pi^{-1}(V)\}$ be a family of subsets of $\mathscr{F}$. Then $(\mathscr{F},\mathcal{O}_{\pi}(\mathscr{F}))$ is a topological space and $\pi:\mathscr{F}\to X$ is a continuous map. For the second condition, let $\mathscr{F}$ be the set of all quantum states on $X$ and $\pi:\mathscr{F}\to X$ be a surjection. Then it is possible to simulate/approximate any quantum state of any quantum theory on $X$.

Given such a triple $(\mathscr{F},\pi,X)$, one can define and run a problem by applying algebraic operators to each $\pi^{-1}(U)$. Let us consider how we can program each $\pi^{-1}(U)$ to solve a problem. To do so, we first decide what kind of quantum state we will address. Since all experimentally observed physical quantities are of finite value, it is sufficient to consider bounded operators. Therefore we use a von Neumann algebra ($W^{*}$-algebra) to embed the problem into $\mathscr{F}$ and write a program. Here a von Neumann algebra is a weakly closed ^∗-algebra of bounded operators on a Hilbert space and contains the identity operator. Note that quantum mechanics includes not only bounded operators but also non-bounded operators. Let $\mathcal{A}(U)$ be a von Neumann algebra on an open set $U$ of $X$. It will act on $\pi^{-1}(U)$ as follows. Suppose a initial quantum state $\rho_{U}\in\mathcal{A}(U)$ is given on $U$. Then one can update it by applying a family of unitary operators $\{a_{\lambda}\}_{\lambda}\subset\mathcal{A}(U)$ to $\rho_{U}$ in such a way that $\rho_{U}\mapsto a_{\lambda}\rho_{U}a^{*}_{U}$. One will be able to simulate a local behavior on an open subset $V\subset U$, by restricting $\mathcal{A}(U)$ to $V$, by which one will obtain a subalgebra $\mathcal{A}(V)\subset\mathcal{A}(U)$. In order to extend $\rho_{U}$ to a state on $W\supset U$, one could do so by embedding $\mathcal{A}(U)$ to $\mathcal{A}(W)$. This extension includes the notion of a tensor product of quantum states. If we can give such an algebraic system to an arbitrary open set of $X$ and reproduce arbitrary quantum states, we can call it a true universal quantum computation. When those operations are defined on a triple $(\mathscr{F},\pi,X)$, we call it quantum fibration (Definition 2). This is clearly a natural extension of conventional quantum computation, which discretely embeds a problem into each fiber $\mathbb{C}^{2^{n}}$ using $n$-qubit Pauli operators.

Another motivation for the author in developing the theory is as follows:

This theory is a well-defined and non-perturbative quantum theory on any topological space.

Conventional quantum theory tries to find a Hamiltonian or Lagrangian a priori to solve a problem, whereas quantum computation finds an algebraic system that can be programmed to reproduce a quantum system to solve a problem. Moreover algebraic operations are well-defined, and if the quantum computation is universal, it can reproduce any state, regardless of non-perturbative/perturbative.

Since quantum computation involves classical computation, we come to the following natural question:

The answer to the above question is given in Definition 3. We propose the notion of semi-classical operations of von Neumann algebra. By restricting each fiber to semi-classical orbits, we would be able to handle semi-classical systems. Quantum and classical states can be distinguished by the presence or absence of quantum correlations such as entanglement. Density operators without entanglement are called separable states and are regarded as classical states. The strength of the entanglement can be measured with what is called an entanglement measure (cf. Example 3.) For example we can define a semi-classical operator by a von Neumann algebra such that it does not increase quantum correlation. In Theorem 3 we verify that any semi-classical von Neumann algebra is indeed closed in classical system.

As some examples of our theory, we address algebraic quantum field theory in Section 5 and quantum chemistry in Section 7.

Acknowledgements. I gratefully acknowledge people at Stony Brook University especially Dmitri Kharzeev, Edward Shuryak, Derek Teaney, Raju Venugopalan, Jacobus Verbaarschot, Ismail Zahed for helpful and stimulating discussions. I thank Pablo Basteiro, Ioannis Matthaiakakis, Rene Meyer at University of Wüerzburg, Adam Lowe and Yoshiyuki Matsuki for useful discussions and collaboration. I also thank Steven Rayan for his encouragement and supervision at University of Saskatchewan. This work was supported in part by Pacific Institute for the Mathematical Science (PIMS) postdoctoral fellowship award and by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Co-design Center for Quantum Advantage (C2QA).

Here, we will admit the existence of fibers that are empty sets. Those empty fibers correspond to the fibers vanished by annihilation operators of particles. The definition of Fibration usually employs the covering homotopy property, but we do not. There are several reasons for this, for example, the homotopy of the fiber is not always readily apparent in any given quantum system. Moreover if $p:E\to X$ is a Serre fibration on a pathwise connected space $X$, then $p^{-1}(x)$ and $p^{-1}(y)$ are homotopy equivalent for all $x,y\in X$. However, general quantum systems do not always satisfy this property. For example, in the most general two-qubit system, pure states are parametrized by $S^{7}$ and a mixed states are parametrized by $SU(4)$, but $S^{7}$ and $SU(4)$ are not homotopy equivalent. General quantum systems with fibers such that a fiber at one point is consists of pure states and a different fiber at another point consists of mixed states cannot be handled by the conventional fibrations.

Now let us consider quantum computation with a quantum fibration which consists of the following data:

1. (1)

   For each open set $U\subset X$, let $\mathcal{A}(U)$ be a von Neumann algebra such that $\mathcal{A}(V)\subset\mathcal{A}(U)$ for all open sets $V\subset U$.

2. (2)

   A density operator $\rho_{U}$ on $U$ is a trace-class operator of $\mathcal{A}(U)$ is defined in such a way that $\rho_{U}=\rho^{*}_{U},\rho_{U}\geq 0,\Tr(\rho_{U})=1$, where $1$ is the identity operator of $\mathcal{A}(U)$.

3. (3)

   There exists a restriction map $r$ such that, for any open set $V\subset U$, $r_{UV}(\rho_{U})$ is a density operator on $V$.

4. (4)

   Let $\Lambda$ be an ordered set. The time-evolution of $\rho_{U}$ is given by a family $\{U_{\lambda}\}_{\lambda\in\Lambda}$ of unitary operator $U_{\lambda}U^{*}_{\lambda}=1$ in such a way that

   (3.1)

   $$\rho_{U}\mapsto U_{\lambda}\rho_{U}U^{*}_{\lambda}\mapsto U_{\lambda^{\prime}}U_{\lambda}\rho_{U}U^{*}_{\lambda}U^{*}_{\lambda^{\prime}}~{}~{}\lambda<\lambda^{\prime}\in\Lambda.$$

The pair $(\rho_{U},\{U_{\lambda}\}_{\lambda})$ generates an orbit

(3.2)

$$O_{\check{\mathcal{A}}(U)}=\{i_{U_{\lambda}}(\rho_{U}):\lambda\in\Lambda\}.$$

There are two ways to define an orbit on each single point $x\in X$. One is simply to chose a von Neumann algebra $\mathcal{A}_{x}$ and construct a von Neumann algebra $\mathcal{A}(U)$ on an open neighber $U$ of $x$ in such a way that $\mathcal{A}_{x}$ is a subalgebra of $\mathcal{A}(U)$. In this case, the restriction map is not given a priori but is defined to be consistent with the algebraic structure on each open subset. Another way to construct an orbit on $x\in X$ is to use the direct limit

(3.3)

$$\mathcal{A}_{x}=\lim_{U\to x}\mathcal{A}(U).$$

In this case, a restriction map and algebraic structures on each open neighbor of $x$ are given a priori.

Once we assign an orbit for each open set $U$ of $X$ with an initial state $\rho_{U}$, we can define a quantum fibration $(\mathscr{F},\pi,X)$ with continuous map

(3.4)

$$\pi:\mathscr{F}\to X$$

such that $\pi^{-1}(U)=O_{\mathcal{A}_{U}}(\rho_{U})$.

To simulate theories on a connected space, it is natural to consider a space in which there are an infinite number of qubits. For example, this would be the case when considering the $N\to\infty$ limit of $\text{SU}(N)$ gauge theory. Let us check that our theory can also address the quantum computation theory with an infinite number of qubits. To this end, let us discuss tensor products of infinite Hilbert spaces. Let $\{\mathcal{H}_{i}\}_{i=1,2,\cdots}$ be a sequence of Hilbert spaces and $\{e_{i}\}_{i=1,2,\cdots}$ be a sequence of their unit vectors $(e_{i}\in\mathcal{H}_{i})$. For each $\bigotimes_{i=1}^{n}\mathcal{H}_{i}$, the embedding map

(3.5)

$$\psi\in\bigotimes_{i=1}^{n}\mathcal{H}_{i}\mapsto\psi\otimes e_{n+1}\in\bigotimes_{i=1}^{n+1}\mathcal{H}_{i}$$

is an isometry map. The direct limit of the direct system $\{\bigotimes_{i=1}^{n}\mathcal{H}_{i}\}_{i=1,2,\cdots}$ defined in this way is a pre-Hilbert space, whose completion with $\{e_{i}\}_{i=1,2,\cdots}$ is called the infinite tensor product of $\{\mathcal{H}_{i}\}_{i=1,2,\cdots}$. We write it as $\bigotimes_{i=1}^{\infty}(\mathcal{H}_{i},e_{i})$. When a von Neumann algebra $\mathcal{A}_{i}$ is given for each Hilbert space $\mathcal{H}_{i}$, one can embed $\widetilde{\mathcal{A}_{n}}=\bigotimes_{i=1}^{n}\mathcal{A}_{i}$ into $\bigotimes_{i=1}^{n+1}\mathcal{A}_{i}$ in such a way that

(3.6)

$$\widetilde{\mathcal{A}_{n}}\ni A\mapsto A\otimes I\in\widetilde{\mathcal{A}_{n}}\otimes\mathcal{A}_{n+1}=\widetilde{\mathcal{A}_{n+1}},$$

by which $\widetilde{\mathcal{A}_{n}}$ is regarded as a subalgebra of $\widetilde{\mathcal{A}_{n+1}}$. This operation is commutative with the embedding (3.5), hence we can embed $\widetilde{\mathcal{A}_{n}}$ into $B\left(\bigotimes_{i=1}^{\infty}(\mathcal{H}_{i},e_{i})\right)$. The von Neumann algebra on $\bigotimes_{i=1}^{\infty}(\mathcal{H}_{i},e_{i})$ determined in this way is called the infinite tensor product

(3.7)

$$\mathcal{A}(\{e_{i}\}_{i})=\bigotimes_{i=1}^{\infty}\mathcal{A}_{i}$$

of $\{\mathcal{A}_{i}\}_{i=1,2,\cdots}$ with respect to $\{e_{i}\}_{i=1,2,\cdots}$. One important remark is that even when each $\mathcal{A}_{i}$ is a set of all complex square matrices $(\mathcal{A}_{i}=M_{m_{i}}(\mathbb{C}))$, there are uncountably many different infinite tensor products of von Neumann algebras and, depending on a choice $\{e_{i}\}_{i=1,2,\cdots}$, $\mathcal{A}(\{e_{i}\}_{i})$ can be type I, II, and III.

All operations in all quantum systems can be viewed as communication channels if we view the initial state as the input state and the final state as the output state. A network is an extension of the communication channels to the entire system. A map sending a quantum state to anther quantum state is called a quantum channel. Let $\mathcal{H}_{1},\mathcal{H}_{2}$ be two Hilbert spaces and $H(\mathcal{H}_{1})=\{\eta\in B(\mathcal{H}_{1}):\eta^{*}=\eta\}$ be the set of all hermitian operators. We call $\Lambda:\text{End}(\mathcal{H}_{1})\to\text{End}(\mathcal{H}_{2})$ a quantum channel if it satisfies the following properties.

• •

  $\Lambda(a\eta_{1}+b\eta_{2})=a\Lambda(\eta_{1})+b\Lambda(\eta_{2})$ for all $\eta_{1},\eta_{2}$ in $H(\mathcal{H}_{1})$ and for all real numbers $a,b$.

• •

  $\Lambda(\eta)\geq 0$ for any $\eta$ in $H(\mathcal{H}_{1})$ such that $\eta\geq 0$.

• •

  $\Tr(\Lambda(\eta))=\Tr\eta$ for any $\eta$ in $H(\mathcal{H}_{1})$.

• •

  For any Hilbert space $\mathcal{H}_{3}$ and for any $\tilde{\eta}$ in $H(\mathcal{H}_{1}\otimes\mathcal{H}_{3})$, if $\tilde{\eta}\geq 0$, then $\Lambda\otimes I(\tilde{\eta})\geq 0$.

To elaborate more on the fourth property of a quantum channel in a general way, let $\mathcal{A}$ be a von Neumann algebra defined on $\mathcal{H}_{1}$. Then the tensor product of $\mathcal{A}$ and $B(\mathcal{H}_{2})$ corresponds to

(3.8)

$$\mathcal{A}\otimes B(\mathcal{H}_{2})=\{a\in B(\mathcal{H}_{1}\otimes\mathcal{H}_{2}):a_{ij}\in\mathcal{A}\},$$

where $a_{ij}$ is defined by equation (4.5). From this, it is straightforward to see that $B(\mathcal{H}_{1})\otimes B(\mathcal{H}_{2})=B(\mathcal{H}_{1}\otimes\mathcal{H}_{2})$. Especially, when $\mathcal{H}_{2}=\mathbb{C}^{n}$, we have $B(\mathcal{H}_{1}\otimes\mathbb{C}^{n})=B(\mathcal{H}_{1})\otimes M_{n}(\mathbb{C})$. Therefore $a\in B(\mathcal{H}_{1}\otimes\mathbb{C}^{n})$ is an $n\times n$ matrix $a=(a_{ij})$ whose components $a_{ij}$ are in $B(\mathcal{H}_{1})$. Thus the condition $\Lambda(\eta)\geq 0~{}(0\leq\eta\in H(\mathcal{H}_{1}))$ is not enough to guarantee $\Lambda\otimes I(\widetilde{\eta})\geq 0~{}(0\leq\widetilde{\eta}\in H(\mathcal{H}_{1}\otimes\mathcal{H}_{2}))$. The physical meaning of this condition is that when there are no interactions between $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$, then events in system $\mathcal{H}_{2}$ do not affect system $\mathcal{H}_{1}$.

Let $\mathscr{F},\mathscr{G}$ be two quantum fibrations on $X$ and $Y$, respectively. Let $\mathcal{O}_{X}$ be the set of all open sets of $X$. For $U\in\mathcal{O}_{X}$ and $V\in\mathcal{O}_{Y}$, let $L(\mathscr{F}_{U},\mathscr{F}_{V})$ denote the set of all quantum channels from $\mathscr{F}_{U}$ to $\mathscr{G}_{V}$. We write

(3.9)

$$L(\mathscr{F},\mathscr{G})=\bigcup_{U\in\mathcal{O}_{X}}\bigcup_{V\in\mathcal{O}_{Y}}\{L(\mathscr{F}_{U},\mathscr{G}_{V})\}$$

and call it quantum network from $\mathscr{F}$ to $\mathscr{G}$. $L(\mathscr{F},\mathscr{G})\cup L(\mathscr{G},\mathscr{F})$ is the bidirectional quantum network between $\mathscr{F}$ and $\mathscr{G}$. In algebraic quantum field theory (AQFT), the assignment $U\to\mathcal{A}(U)$ for each $U\in\mathcal{O}_{X}$ is called a net of local von Neumann algebras. Defining channels among local algebras allows them to interact and communicate, by which we mean it a network of local von Neumann algebras.

To apply our theory to semi-classical and classical phenomena, we consider a semi-classical class of operators.

Let $f:\widetilde{D}(\mathcal{H})\to\mathbb{R}_{\geq 0}$ be a non-negative map. Let

(6.1)

$$D_{0}(f)=\{\rho\in\widetilde{D}(\mathcal{H}):f(\rho)=0\}$$

be the set of all elements of $\widetilde{D}(\mathcal{H})$ sent to 0 by $f$. For a given von Neumann algebra $\mathcal{A}$, we define its subset as

(6.2)

$$\mathcal{A}^{f}=\left\{a\in\mathcal{A}:\forall\rho\in\widetilde{D}(\mathcal{H}),~{}i_{a}(\rho)\in\widetilde{D}(\mathcal{H}),~{}f\left(\frac{i_{a}(\rho)}{\Tr(i_{a}(\rho))}\right)\leq f\left(\frac{\rho}{\Tr\rho}\right)\right\}.$$

We put

(6.3)

$$\mathcal{C}\left(\mathcal{D}_{0}(f),\mathcal{A}^{f},\mathcal{H}\right)=\bigcup_{\rho\in{D_{0}(f)}}O_{\mathcal{A}^{f}}(\rho)$$

The following statement plays a fundamental role for discussing classical states and classical operators (Definition 3).