This article presents a sequential growth model for the Universe that acts like a quantum computer. The basic constituents of the model are a special type of causal set (causet) called a c-causet. A c-causet is defined to be a causet that has a unique labeling. We characterize c-causets as those causets that form a multipartite graph or equivalently those causets whose elements are comparable whenever their heights are different. We show that a c-causet has precisely two c-causet offspring. It follows that there are c-causets of cardinality . This enables us to classify c-causets of cardinality in terms of n-bits. We then quantize the model by introducing a quantum sequential growth process. This is accomplished by replacing the n-bits by n-qubits and defining transition amplitudes for the growth transitions. We mainly consider two types of processes, called stationary and completely stationary. We show that for stationary processes, the probability operators are tensor products of positive rank-one qubit operators. Moreover, the converse of this result holds. Simplifications occur for completely stationary processes. We close with examples of precluded events.
One frequently hears people say that the Universe acts like a giant quantum computer, but when pressed, they are usually short on the details. This article attempts to begin giving these details. It should be emphasized that only a basic framework is presented, and much work remains to be done. If this idea is correct, then great benefits will result. One benefit is the better understanding of the Universe itself, and another is the ability to greatly increase our computational power.
We first present a theory of discrete quantum gravity in terms of causal sets (causets) [1,2,3]. Unlike previous sequential growth models, the basic elements of this theory are a special type of causet called a covariant causet (c-causet). A c-causet is defined to be a causet that has a unique labeling. That is, two different labelings of a c-causet are isomorphic. The restriction of a growth model to c-causets provides great simplifications. For example, every c-causet possesses a unique c-causet history and has precisely two covariant offspring. It follows that there are c-causets of cardinality . This enables us to classify c-causets of cardinality in terms of n-bits. The framework of a classical computer is already emerging. We characterize c-causets as those causets that form a multipartite graph or equivalently those causets whose elements are comparable whenever their heights are different.
We next quantize the model by introducing a quantum sequential growth process. This is accomplished by replacing the n-bits with n-qubits and defining transition amplitudes for the growth transitions. The transition amplitudes are given by complex-valued coupling constants , . If the coupling constants are independent of j, we call the process stationary, and if they are independent of n and j, we call the process completely stationary. We show that for stationary processes, the probability operators that determine the quantum dynamics are tensor products of rank-one qubit operators. Moreover, the converse of this result holds. Simplifications occur for completely stationary processes. In this case, all of the qubit operators are the same and can be related to spin operators. We close with some examples of precluded events in the completely stationary case.
2. Covariant Causets
In this article, we call a finite partially ordered set a causet. If two causets are order isomorphic, we consider them to be identical. If a and b are elements of a causet x, we interpret the order as meaning that b is in the causal future of a and a is in the causal past of b. An element is maximal if there is no with . If and there is no with , then a is a parent of b and b is a child of a. If , we say that a and b are comparable if or . A chain in x is a set of mutually comparable elements of x, and an antichain is a set of mutually incomparable elements of x. The height of is the cardinality of the longest chain whose largest element is a. The height of x is the maximum of the heights of its elements. We denote the cardinality of x by .
If x and y are causets with , then x produces y if y is obtained from x by adjoining a single maximal element a to x. In this case, we write and use the notation . If , we also say that x is a producer of y and y is an offspring of x. In general, x may produce many offspring, and y may be the offspring of many producers.
A labeling for a causet x is a bijection , such that with implies that . A labeled causet is a pair , where ℓ is a labeling of x. For simplicity, we frequently write and call x an ℓ-causet. Two ℓ-causets x and y are isomorphic if there exists a bijection , such that if and only if and for every . Isomorphic ℓ-causets are considered identical as ℓ-causets. It is not hard to show that any causet can be labeled in many different ways, but there are exceptions, and these are the ones of importance in this work. A causet is covariant if it has a unique labeling (up to ℓ-causet isomorphism). Covariance is a strong restriction, which says that the elements of the causet have a unique “birth order” up to isomorphism. We call a covariant causet a c-coset.
We denote the set of c-causets with cardinality n by and the set of all c-causets by . Notice that any nonempty c-causet y has a unique producer. Indeed, if y had two different producers , then and could be labeled differently, and these could be used to give different labelings for y. If , then the parent-child relation makes x into a graph . A graph G is multipartite if there is a partition of its vertices , such that the vertices of and are adjacent and there are no other adjacencies.
Theorem 2.1.The following statements for a causet x are equivalent. (a) x is covariant, (b) the graph is multipartite, (c) are comparable whenever a and b have different heights.
Proof. Conditions (b) and (c) are clearly equivalent. To prove that (a) implies (b), suppose x is covariant and let , where is the set of elements in x of height i. Suppose , and . We can delete maximal elements of y until b is maximal and the only element of height . Denote the resulting causet by z. We can label b by , a by and consistently label the other elements of z, so that z is an ℓ-causet. We can also label b by , a by and keep the same labels for the other elements of z. This gives two nonisomorphic labelings of z. Adjoining maximal elements to z to obtain x, we have x with two nonisomorphic labelings, which is a contradiction. Hence, so a is a parent of b. It follows that x is multipartite. To prove that (b) implies (a), suppose the graph is multipartite. Letting where is the set of elements of height i, it follows that for all , , . We can write:
where j is the label on . This gives a labeling of x and is the only labeling up to isomorphism. ☐
Theorem 2.2.If , then x has precisely two covariant offspring.
Proof. By Theorem 2.1, the graph is multipartite. Suppose x has height n. Let where a has all of the elements of height n as parents. Then, a is the only element of with height . Hence, is multipartite, so by Theorem 2.1, is a covariant offspring of x. Let , where b has all of the elements of height in x as parents (if , then b has no parents). It is clear that is a multipartite graph. By Theorem 2.1, is a covariant offspring of x. Furthermore, there is only one covariant offspring of each of these two types. Let be a covariant offspring of x that is not one of these two types, and let have label . Then, a and c are incomparable, and we can label c by . If we interchange the labels of a and c, we get a nonisomorphic labeling of y, which gives a contradiction. We conclude that x has precisely two covariant offspring. ☐
Corollary 2.3.There are c-causets of cardinality .
Proof. Notice that we obtain all c-causets from the producer-offspring process of Theorem 2.2. Indeed, take any and delete maximal elements until we arrive at the one element c-causet. In this way, x is obtained from the process of Theorem 2.2. We now employ induction on n. There are c-causets of cardinality one. If the result holds for c-causets of cardinality n, then by Theorem 2.2, there are c-causets of cardinality . Hence, the result holds for c-causets of cardinality . ☐
As a bonus, we obtain an already known combinatorial identity. A composition of a positive integer n is a sequence of positive integers whose sum is n. The order of terms in the sequence is taken into account. For example, the following are the compositions of .
The reader has surely noticed that for , the number of compositions of n is .
Corollary 2.4.There are compositions of the positive integer n.
Proof. There is a bijection between compositions of n and multipartite graphs with n vertices. The result follows from Corollary 2.3. ☐
The pair forms a partially ordered set in its own right. Moreover, also forms a graph that is a tree. Figure 1 depicts the first five levels of this tree. The binary designations in Figure 1 will now be explained. By Corollary 2.3, at height , there are c-causets, so binary numbers fit well, but how do we define a natural order for the c-causets? We have seen in Theorem 2.2 that if , , then x has precisely two offspring in , , where has the same height as x and has the height of x plus one. We call the 0-offspring and the 1-offspring of x. We assign a binary order to recursively as follows. If , then x is the unique one element c-causet, and we designate x by 0. If , then x has a unique producer . Suppose y has binary order , or 1. If x is the 0-offspring of y, then we designate x with , and if x is a 1-offspring of y, then we designate x with . The reader can now check this definition with the binary order in Figure 1.
We now see the beginning development of a giant classical computer. At the -th step of the process, n-bit strings are generated. It is estimated that we are now at about the -th step, so -bit strings are being generated. There are about such strings, so an enormous amount of information is being processed. When we get to quantum computers, then superpositions of strings will be possible, and the amount of information increases exponentially. It is convenient to employ the notation:
for an n-bit string. In this way, we can designate each uniquely by , where . For example, the c-causets at Step 3 in Figure 1 are . In decimal notation, we can also write these as .
Five Steps of a Multiverse
Five Steps of a Multiverse
The binary order that we have just discussed is equivalent to a natural order in terms of the c-causet structure. Let , where we can assume without loss of generality that j is the label of , . Define:
Thus, is the set of labels of the descendants of . Order the set of c-cosets in lexicographically as follows. If , then if:
It is easy to check that < is a total order relation on . The next theorem, whose proof we leave to the reader, shows that the order < on is equivalent to the binary order previously discussed.
Theorem 2.5.If , then if and only if .
Example 1. We can illustrate Theorem 2.5 by considering . For the c-causets , we list the sets . Notice that we need not list in all cases of .
The lexicographical order becomes:
Example 2. This is so much fun, that we list the sets:
for the c-causets .
This order structure induces a topology on , whereby we can describe the “closeness” of c-causets. For example, we can place a metric on by defining . If we want to keep the size of the metric reasonable, we could define:
3. Quantum Sequential Growth Processes
The tree can be thought of as a growth model, and an is a possible universe at step (time) n. An instantaneous universe x grows one element at a time in one of two ways at each step. A path in is a sequence (string) where and . An n-path is a finite sequence , where, again, and . We denote the set of paths by Ω and the set of n-paths by . We think of as a “completed” universe or as a universal history. We may also view as an evolving universe. Since a c-causet has a unique producer, an n-path is completely determined by . In other words, a c-causet possesses a unique history. We can thus identify with , and we write . If , we denote by the two element subset of consisting of , where and are the offspring of . Thus,
If , we define by:
Thus, is the set of one-element continuations of n-paths in A.
The set of all paths beginning with is called an elementary cylinder set and is denoted by . If , then the cylinder set is defined by:
Using the notation:
we see that:
is an increasing sequence of subalgebras of the cylinder algebra . Letting be the σ-algebra generated by , we have that is a measurable space. For , we define the sets by:
That is, is the set of n-paths that can be continued to a path in A. We think of as the n-step approximation to A. We have that:
so that . However, in general, even if .
Let be the n-path Hilbert space with the usual inner product:
For , the characteristic function has norm . In particular, satisfies:
A positive operator ρ on that satisfies is called a probability operator . Corresponding to a probability operator ρ, we define the decoherence functional [1,4,5]:
by . We interpret as a measure of the interference between the events A and B when the system is described by ρ. We also define the q-measure by and interpret as the quantum propensity of the event [1,3,6]. In general, is not additive on , so is not a measure. However, is Grade-2 additive [1,3,6] in the sense that if are mutually disjoint, then:
Let be a probability operator on , . We say that the sequence is consistent if:
for all . We call a consistent sequence a covariant quantum sequential growth process (CQSGP). Let be a CQSGP and denote the corresponding q-measure by . A set is suitable if exists (and is finite), in which case, we define . We denote the collection of suitable sets by . Of course, with , . If and , where , then it follows from consistency that . Hence, and . We conclude that , and it can be shown that the inclusions are proper, in general. In a sense, μ is a q-measure on that extends the q-measures .
There are physically relevant sets that are not in . In this case, it is important to know whether such a set A is in and, if it is, to find . For example, if , then:
but . As another example, the complement . Even if , since for in general, it does not follow immediately that . For this reason, we would have to treat as a separate case.
We saw in Section 2 that we can represent each element of uniquely as , where and can be considered as a binary number. We can also represent each element in as a n-bit binary number , or 1. Since , we can also represent each by an n-bit binary number . The standard basis for is the set of vectors , . We frequently use the notation , which is called the computational basis in quantum computation theory. In this theory, is represented by:
where is or , which form the basis of the two-dimensional Hilbert space .
The basis vectors and are called qubit states, but we shall call them qubits, for short. We also call , given above, an n-qubit. This is the quantum computation analogue of an n-bit of classical computer science. If is a probability operator, the corresponding decoherence matrix is the complex matrix, whose component is given by:
This is frequently shortened to:
but we shall not use this notation, because it can be confusing. For , we form the superpositions:
The decoherence functional is now given by:
Superpositions are a strictly quantum phenomenon that has no counterpart in classical computation.
An event is precluded if . Precluded events have been extensively studied in [2,3,4,7,8], and they are considered to be events that never occur. We shall give simple examples later that show that if A is precluded and , then B need not be precluded. However, the following properties do hold.
Theorem 3.1. (a) If is precluded and is disjoint from A, then . (b) If are disjoint precluded events, then is precluded.
Proof. (a) Since , we have that:
Hence, so . Since, , we have that:
Part (b) follows from (a). ☐
An event is precluded if and A is strongly precluded if there exists an , such that for all . For example, if , where and , then A is strongly precluded. Of course, strongly precluded events are precluded.
A precluded event is primitive if it has no proper, nonempty precluded subsets.
Theorem 3.2.If is precluded, then A is primitive or A is a union of mutually disjoint primitive precluded events.
Proof. If A is primitive, we are finished. Otherwise, there exists a proper, nonempty precluded subset . Since , there exists a nonempty, primitive precluded event . Applying Theorem 3.1, we conclude that . In a similar way, there exists a nonempty, primitive precluded event . Of course, . Continuing, this process must eventually stop, and we obtain a sequence of mutually disjoint primitive preluded events with . ☐
4. Covariant Amplitude Processes
This section considers a method of constructing a CQSGP called a covariant amplitude process. Not all CQSGPs can be constructed in this way, but this method appears to have physical motivation .
A transition amplitude is a map , such that if and for all . This is similar to a Markov chain, except may be complex. The covariant amplitude process (CAP) corresponding to is given by the maps where:
We can consider to be a vector in . Notice that for , we can define to be , where is the unique history of x. Observe that:
and we also have that:
Define the rank-one positive operator on . The norm of is:
Since , we conclude that is a probability operator. It is shown in  that is consistent, so forms a CQSGP. We call the CQSGP generated by the CAP .
The decoherence functional corresponding to the CAP becomes:
In particular, for , the decoherence matrix elements:
are the matrix elements of in the standard basis. The q-measure is given by:
In particular, for every and . Of course, we also have that for all .
Since each has precisely two offspring, we can describe a transition amplitude and the corresponding CAP in a simple way. Let:
. We call the numbers coupling constants for the corresponding CAP .
Example 3. If the CAP has coupling constants , then we have , , , , , .
We shall only need a special case of the next theorem, but it still has independent interest.
Theorem 4.1.An operator M on is a rank-one probability operator if and only if M has a matrix representation , where , , satisfy .
Proof. Suppose with , where . Let be the vector . We have that , so M is positive with rank-one. To show that M is a probability operator, we have:
Conversely, let M be a rank-one probability operator. Since M is rank-one, it has the form for some . We then have the matrix representation:
where , , is the standard basis for . Letting , we conclude that . Since M is a probability operator, we have that:
Now, there exists a , such that . Letting , we obtain:
where , . Hence, . ☐
An operator on is called a qubit operator. We shall only need the following corollary of Theorem 4.1.
Corollary 4.2.A qubit operator M is a rank-one probability operator if and only if M has a matrix representation:
A CAP is stationary if the coupling constants are independent of j. In this case, we write , and we have . By Corollary 4.2, the operators:
are qubit rank-one probability operators.
Theorem 4.3.Let be the coupling constants for a stationary CAP. The generated CQSGP has the form:
Proof. Since , we can write:
At the next step, we apply Example 3 to obtain:
Continuing by induction, we have that (4.2) holds.
Equation (4.2) shows that the -qubit probability operator is the tensor product of qubit probability operators. The next result shows that the converse of Theorem 4.3 holds.
Theorem 4.4.If the CQSGP has the form:
where is a rank-one probability operator, then is generated by a stationary CAP.
Proof. Since , , is a rank-one qubit probability operator, by Corollary 4.2, we have that:
where . As in the proof of Theorem 4.3, is generated by a stationary CAP whose coupling constants are . ☐
We say that a CAP is completely stationary if the coupling constants are independent of n and . In this case, we have a single coupling constant , and the generated CQSGP has the form:
In this section, we compute some simple examples of q-measures in the stationary case. Let be a stationary CAP with corresponding coupling constants . As usual, we can identify with . If , we have that . For , we have , , so and . For:
we have , , , . Hence, , , and . We now compute the q-measure of some two element sets. We have that:
Since in general, we conclude that and interfere with each other, except in special cases. If:
we say that and interfere destructively, and if:
we say that and interfere constructively. The three possible cases, , can occur depending on the value of . In a similar way, we have that :
It follows that any pair of elements of interfere, in general. Finally, we compute the q-measures of some three element sets:
We now consider:
In this case, we have , , , , , , , . We then have that , . In general, the pattern is clear that:
where if the history of turns “left” at the i-th step and if it turns “right” at the i-th step. Some q-measures of two element sets are:
In general, any pair of c-causets in interfere.
We now consider the extremal left path . Is ? We have that:
Now, if and only if exists, and this depends on the values of . In fact, we can set values of so that for an . For example, if we let , then we obtain:
Moreover, in this case, for every with similar values for .
As another example, let be the set of paths , such that are the “middle half” of . That is, , , ,
Now, , :
It is not a coincidence that . In fact, and . It follows that , so with . In a similar way, with . We can interpret as the “one fourth end paths” with , .
The situation for non-cylinder sets is more complicated, so to simplify matters, we consider a completely stationary CAP. In this case, we have only one coupling constant c. For , we have that , where , j is the number of “left turns” and k is the number of “right turns.” We then have explicitly that:
The q-measure of becomes:
It is interesting that in this case, we have:
If , then:
Whether exists or not depends on c. If , then for every and . If , then for every . If , or vice versa, then for some and for others. Except for the trivial cases or , we have that whenever . An interesting example of a set is:
Thus, where is the extremal left path. Then, and . If , then so with .
As a special case, let be a completely stationary CAP with coupling constant . This is probably the simplest nontrivial coupling constant. Notice that and . Moreover:
For , we have that . It follows that for every and . In a similar way, if is finite, then and . Moreover, and . In , we have that:
so, in this case, and do not interfere. In a similar way, , so and do not interfere. On the other hand,
so and interfere destructively. Furthermore,
so and interfere constructively. Even in this simple case, we can get strange results:
We can check Grade-2 additivity:
An interesting property of this special case is that the probability operators are closely related to the Pauli spin operator:
In particular, for , we have:
In this way, corresponds to a state for spin- particles.
We now consider precluded events for the CAP that we are discussing. We say that are an antipodal pair if . Since , , we have that:
for some . It follows that and are an antipodal pair if and only if:
for some . We leave the proof of the following result to the reader. As usual, we apply the identity .
Theorem 5.1.A set is a nonempty, primitive precluded event if and only if , where and are an antipodal pair.
Applying Theorems 5.1 and 3.2, we obtain:
Corollary 5.2.A set is precluded if and only if A is a disjoint union of antipodal pairs.
Example 4. We illustrate Corollary 5.2 by displaying the antipodal pairs in , and . In , there is only one antipodal pair . In , the antipodal pairs are:
In , there are 28 antipodal pairs. To save writing, we use the notation . The antipodal pairs in are:
According to the coevent formulation [3,5,7,8], precluded events do not occur, so we can remove them from consideration. What is left can occur in some a homomorphic realization of possible universes [3,7,8]. We can remove a precluded event from (or ), which is as large as possible, but there is no unique way of doing this, in general. To illustrate this method, let us remove the “left” and “right” precluded extremes. In , we remove the precluded event , and we obtain:
with . In , we remove the precluded event:
and we obtain:
with . In , we remove the precluded event:
and we obtain:
with . Continuing this process, we conjecture that we obtain a sequence of events , where and . Although increases exponentially, if this conjecture holds, then only increases linearly. This gives a huge reduction for the number of possible universes. If satisfies , then with and with . We would then conclude that is precluded, and a realizable universe would have to be in A.
6. Concluding Remarks
Let us emphasize that we did not say that the Universe is a giant quantum computer. We are only suggesting that the present mathematical model for the Universe begins to resemble a quantum computer. Moreover, we do not mean that employing this model will eventually enable us to tap into the Universe with our laptops to gain enormous computational power. Such a goal may actually be impossible, because we have presented a multiverse theory and communication between individual universes will probably not be possible. What will be attainable is the design of quantum computers at the nuclear or even elementary particle level with a subsequent huge increase in speed and memory. In fact, the c-causets we have presented describe a framework or scaffolding for the universe. Each vertex of a c-causet may represent a four-dimensional cell of Planckian volume. We may be able to take advantage of this granular space-time structure to build quantum computers.
The precise definition of a quantum computer is fairly technical, and we do not need to consider it here. The rough idea is that two or more particles are prepared in an initial quantum state (frequently entangled) and are directed through a sequence of quantum gates described by unitary operators. The gates are designed to implement a quantum algorithm for performing a desired computation. The resulting final state is subjected to a measurement called a quantum operation. The measurement value gives information about the computation. The process may be repeated a number of times to gain more information. The important point for us is that the quantum states are usually constructed from two-level quantum systems, such as electron spin in a fixed direction or photon polarization. One promising possibility is nuclear spin in a nuclear magnetic resonance machine. Such two-level systems are called qubits. By combining n of these systems (say n electrons), we obtain an n-qubit whose states are superpositions of tensor products of n qubit states. We have seen in Section 3 that n-qubits also describe c-causets corresponding to various universes. Although the universes do not evolve exactly according to a unitary dynamics (for one thing, the corresponding Hilbert spaces have increasing dimensions), as we saw in Section 4, the dynamics is governed by a transition amplitude. These amplitudes are characteristic of quantum mechanics, and it is well-known that they are closely related to unitary operators. In a rough sense, the universes evolve by passing through quantum gates. Of course, the Universe does not itself make a conscious measurement. This is up to us, the observers. In summary, this is why we say that our model begins to resemble a quantum computer.
We have one final remark. It is possible that the concept of a c-causet is too strong, so our discrete quantum gravity model is too restrictive to describe a universal sequential growth process. If this is the case, we can still say that a subsystem of the full system begins to behave like a quantum computer. At the very least, certain regular quantum systems may well be used as computer memory to store vast amounts of information.
Conflicts of Interest
The authors declare no conflict of interest.
Gudder, S. An approach to discrete quantum gravity. ArXiv E-Prints2013. [Google Scholar]
Sorkin, R. Causal sets: Discrete gravity. In Lectures on Quantum Gravity; Springer: New York, NY, USA, 2005. [Google Scholar]
Surya, S. Directions in causal set quantum gravity. ArXiv E-Prints2011. [Google Scholar]