A quantum computing-based approach to improve the efficiency of library information retrieval

Quantum Bayesian networks based on density operators are to be based on the following four assumptions. First, many identical quantum states are needed as a large data set in the first place. Second, there is no noise or coupling to the unknown external environment when the quantum state receives the quantum operator. Third, there is no feedback in the system. Then, the definition of the inputs and outputs of each operator is obvious, and the graph is acyclic, i.e., it is a directed acyclic graph. Fourth, the 1qubit in both quantum systems merges into a joint state by tensor product. Similarly, the 2qubit quantum state splits into two 1qubit quantum states is by tracking the particle paths in each dimension. That is, if they merge or separate the operation destroys the quantum entangled state.

There are four kinds of operation operators in the quantum probabilistic graphical model: the measurement operator, the you operator, the merger operator and the separation operator. Their representation is shown in Fig. 2.

Figure 2.

The operation operators of quantum probabilistic graphical models

The joint probability of the quantum probabilistic graphical model is determined by all the quantum states and measurements. The probability of output n→{\vec n} under the condition of hidden state ρ with measurement operator M shown in Fig. 2(a) is equation (2): 2P(n→∣M,ρ)=∏i=1d〈φi|ρφi|niP(\vec n\mid M,\rho ) = \prod\limits_{i = 1}^d {\langle {\varphi _i}|} \rho \left. {{{\left. {{\varphi _i}} \right|}^{{n_i}}}} \right\rangle where { φi }i=1d\left\{ {{\varphi _i}} \right\}_{i = 1}^d includes a series of orthogonal bases of M and n_i, and n_i is the ith component of vector n→{\vec n}.

The density matrices of the input and output of the You operator are determined. Figure 2(b) Then the conditional probability of the output You matrix can be expressed as equation (3): 3P(ρ2∣U,ρ1)=δ(ρ2,Uρ1U+)P\left( {{\rho _2}\mid U,{\rho _1}} \right) = \delta \left( {{\rho _2},U{\rho _1}{U^ + }} \right) where if x₁ = x₂, then δ(x₁, x₂) = 1; otherwise δ(x₁, x₂) = 0.

A merger operator is the transformation of two quantum states in lower dimensions into one quantum state in higher dimensions. According to Assumption IV, the uncoupled joint state is the tensor product of the two inputs. As in Fig. 2(c), the conditional probability can also be expressed as Eq. (4): 4P(ρ12∣MS,ρ1,ρ2)=δ(ρ12,ρ1⊗ρ2)P\left( {{\rho _{12}}\mid MS,{\rho _1},{\rho _2}} \right) = \delta \left( {{\rho _{12}},{\rho _1} \otimes {\rho _2}} \right)

Conversely, a separation operator divides a quantum state into two states. According to Assumption IV and the opposite operation of the tensor product, the density matrix of each quantum state is the trace of the joint density matrix particle, which is a subspace of the other edge state. The conditional probability of Fig. 2(d) is represented as Eqs. (5)-(6): 5P(ρ1∣MS,ρ12)=δ(ρ1,tr2(ρ12))P\left( {{\rho _1}\mid MS,{\rho _{12}}} \right) = \delta \left( {{\rho _1},{{{\mathop{\rm tr}\nolimits} }_2}\left( {{\rho _{12}}} \right)} \right) 6P(ρ2∣MS,ρ12)=δ(ρ2,tr1(ρ12))P\left( {{\rho _2}\mid MS,{\rho _{12}}} \right) = \delta \left( {{\rho _2},{{{\mathop{\rm tr}\nolimits} }_1}\left( {{\rho _{12}}} \right)} \right)

From Eq. (4)-Eq. (6), the joint probability of the quantum probabilistic graphical model is given by Eq. (7): 7L=∏U∈VUδ(ρUout ,UρUin U+)⋅∏MS∈VMS1δ(ρMS12,ρMS1⊗ρMS2)⋅∏MS∈VMS2δ(ρMS1,tr2(ρMS12))δ(ρMS2,tr1(ρMS12))⋅∏M∈VM∏i〈φMi|ρM|φMi〉nMi\matrix{ L \hfill & = \hfill & {\prod\limits_{} {_{U \in {V_U}}\delta } \left( {{\rho _{{U_{{\rm{out }}}}}},U{\rho _{{U_{{\rm{in }}}}}}{U^ + }} \right)} \hfill \cr {} \hfill & {} \hfill & { \cdot \prod\limits_{} {_{MS \in {V_{M{S_1}}}}\delta } \left( {{\rho _{M{S_{12}}}},{\rho _{M{S_1}}} \otimes {\rho _{M{S_2}}}} \right)} \hfill \cr {} \hfill & {} \hfill & { \cdot \prod\limits_{} {_{MS \in {V_{M{S_2}}}}\delta } \left( {{\rho _{M{S_1}}},t{r_2}\left( {{\rho _{M{S_{12}}}}} \right)} \right)\delta \left( {{\rho _{M{S_2}}},t{r_1}\left( {{\rho _{M{S_{12}}}}} \right)} \right)} \hfill \cr {} \hfill & {} \hfill & { \cdot \prod\limits_{} {_{M \in {V_M}}\prod\limits_{} {_i\langle {\varphi _{{M_i}}}|} } {\rho _M}{{\left| {{\varphi _{{M_i}}}} \right\rangle }^{{n_{{M_i}}}}}} \hfill \cr } where ρ_Uin and ρ_Uout denote the density matrices of the input and output You operators U, V_MS1 and V_MS1 are the sets of separation and merger operators, ρ_MS12 is the joint state, ρ_MS1 and ρ_MS2 are the corresponding edge states, ρ_M is the measurement operator M of the input quantum state, n_Mi is the ith output of the measurement operator M, and φ_Mi is the ith basis of the measurement operator M.

Based on the principle of probabilistic reconstruction, graphical models have a corresponding quantum graph model. Two examples are briefly described below. Since quantum probabilistic graphical models mainly include quantum Bayesian models and quantum Markov random fields (QMRF).

(1) A binary measurement of a quantum state: a quantum Bayesian network is shown in Fig. 3, where Fig. 3(a) is the simplest quantum Bayesian network: a measurement of a 1 qbit quantum state. The input is the density matrix of the measurement operator M and the output vector n→=(n0,n1)\vec n = \left( {{n_0},{n_1}} \right) is the result of the binary measurement. M includes two orthogonal bases |φ₀〉 and |φ₁〉.

Figure 3.

Quantum Bayesian network

(2) Quantum Hidden Markov Model (QHMM): a quantum hidden Markov model is a type of quantum Bayesian network, all directed acyclic graphs. Unlike classical systems, it is measured using a set of basis vectors, but is not sufficient to determine a quantum state. In order to adequately characterize a quantum state, it is represented by a series of cascaded You operators. This is the quantum Markov model shown in Fig. 3(b). Its likelihood function is equation (8): 8L=∏k| φkρ1 |φk〉〉⋅[ ∏lδ(ρl+1,UlρlUl+)⋅∏k|φk〉ρl+1|φk〉nl+1,k ]\left. {\left. {L = \prod\limits_k {\left| {{\varphi _k}{\rho _1}} \right|} {\varphi _k}} \right\rangle } \right\rangle \cdot \left[ {\prod\limits_l \delta \left( {{\rho _{l + 1}},{U_l}{\rho _l}U_l^ + } \right) \cdot \prod\limits_k {\left| {{\varphi _k}} \right\rangle } {\rho _{l + 1}}{{\left| {{\varphi _k}} \right\rangle }^{{n_{l + 1,k}}}}} \right]

A dual term layer is selected to reflect the associations within the term nodes. Let R and R_i denote the original term layer as well as the term nodes respectively, all term nodes R_i existing within the original term layer R are copied, and the acquired term nodes Ri′R_i^\prime are used to build a new term layer, denoted by R′. The pointing of arcs between term nodes within different layers is obtained using ontology-based association between term nodes, which is performed as follows:

(1) Setting the correlation between term R_i and term Ri′R_i^\prime to 1 gives Ri′→RiR_i^\prime \to {R_i}, which means that the arc moment of term Ri′R_i^\prime pointing to term R_i exists.

(2) When there is an ontology correlator relationship between term R_i and term Rj, Srd(R_i, R_j) is used to denote the ontology correlation between the terms, and the correlation between the two is set to be greater than a certain threshold, at which time Ri′→RjR_i^\prime \to {R_j}, Rj′→RiR_j^\prime \to {R_i}, i.e., the arc of term Ri′R_i^\prime pointing to term R_j, and the arc of term Rj′R_j^\prime pointing to term R_i, exist at the same time.

The set of parent nodes of term R_i is denoted by pa(R_i). The set of values of the term variable Ri′R_i^\prime , which exists within the term layer R′, is { r¯′,r′ }\left\{ {{{\bar r}^\prime },{r^\prime }} \right\}, which is a term binary random variable, where it is denoted by r¯′{{\bar r}^\prime } when the terms are irrelevant and by r′ when the terms are relevant.

Let Ti′T_i^\prime be a randomly existing root term node, the edge probability associated with this root term node needs to be clarified, and by setting the probability of all term nodes within a given set to be the same, the edge probability associated with the root term node can be obtained as equation (9): 9P(r′)=1/MP\left( {{r^\prime }} \right) = 1/M where M denotes the total number of term nodes in the set.

The root term node irrelevance probability is as in equation (10): 10P(r¯i′)=1−P(ri′)P\left( {\bar r_i^\prime } \right) = 1 - P\left( {r_i^\prime } \right)

The parent node of the node within the Bayesian network determines the probability of the random non-root node, let R_i be the random non-root term node within the set, and the combination of correlated and uncorrelated values of the term variables within pa(R_i) is also represented by pa(R_i). In this way, we obtain the general regular model probability function as in Eq. (11): 11P(ri∣pa(Ri))=∑Rj′∈pa(Ri),rj∈pa(Ri)vijP\left( {{r_i}\mid pa\left( {{R_i}} \right)} \right) = \sum\limits_{{R_{{j^\prime }}} \in pa\left( {{R_i}} \right),{r_j} \in pa\left( {{R_i}} \right)} {{v_{ij}}} where v_ij denotes the weight of term Rj′R_j^\prime affecting term R_i.

When there are many parent nodes for term R_i, the weight v_ij can be obtained as equation (12): 12vij={ η,0.5≤η≤1.0,i=j1−ηMaxSrdSrd(Ri,Rj′),i≠j{v_{ij}} = \left\{ {\matrix{ {\eta ,0.5 \le \eta \le 1.0,i = j} \hfill \cr {{{1 - \eta } \over {MaxSrd}}Srd\left( {{R_i},R_j^\prime } \right),i \ne j} \hfill \cr } } \right. where η and Srd denote the regulation parameters and the sum of term ontology associations within the set of term nodes, respectively.

The maximum value of the sum of term ontology correlations is equation (13): 13MaxSrd=max(∑Rj∈pa(Ri)Srd(Ri,Rj′))MaxSrd = max\left( {\sum\limits_{{R_j} \in pa\left( {{R_i}} \right)} {Srd} \left( {{R_i},R_j^\prime } \right)} \right)

The sum of the effects of term-related terms on the term is less than the effect of the term on itself, which is evident when i = j, 0.5 ≤ η ≤ 1.0.

Let B_j denote the document that exists within the set to get its conditional probability as equation (14): 14P(bj∣pa(Bj))=∑Ri∈Bj,ri∈pa(Bj)wijP\left( {{b_j}\mid pa\left( {{B_j}} \right)} \right) = \sum\limits_{{R_i} \in {B_j},{r_i} \in pa\left( {{B_j}} \right)} {{w_{ij}}} where pa(B_j) and w_ij denote the combination of relevant and irrelevant values of each term variable in pa(B_j) and the weight of index term R_j of document B_j, respectively. The above formulas need to satisfy w_ij 0(∀i, j), ∑_Ri∈Bj W_ij ≤ 1(∀_j). When r_i ∈ pa(B_j), it means the sum of the weights of the related terms in pa(B_j).

The correlation probability value of B_j is higher when there are more correlated terms in pa(B_j) the document. The TF-IDF algorithm is selected to calculate w_ij as in equation (15): 15wij=γ−1rfij×ibfi2∑Rk∈Bjrfkj×ibfk2{w_{ij}} = {\gamma ^{ - 1}}{{r{f_{ij}} \times ibf_i^2} \over {\sqrt {\sum\limits_{{R_k} \in {B_j}} r {f_{kj}} \times ibf_k^2} }} where γ is a specification constant at ∑_Ri∈Bj W_ij ≤ 1 and satisfies ∀B_j ∈ B, rf_ij and ibf_i denote term frequency and backward document frequency, respectively.

Let Q be the user query as well as the submitted information, and the relevance P(B_j|Q) denotes the conditional probability of obtaining the document B_j at the time of query Q. The steps to obtain the relevance are as follows:

(1) The edge probability belonging to the term Q is instantiated when the user submits the query information Q. When Rj′∈QR_j^\prime \in Q as well as Rj′∉QR_j^\prime \notin Q, the results are obtained as P(Rj′∣Q)=1P\left( {R_j^\prime \mid Q} \right) = 1 and P(Rj′∣Q)=1/MP\left( {R_j^\prime \mid Q} \right) = 1/M, respectively.

(2) The a posteriori probability of a random term R_i within a term layer R is obtained based on equation (16): 16P(Ri∣Q)=∑Rj∈pa(Ri)vijP(Rj′∣Q)P\left( {{R_i}\mid Q} \right) = \sum\limits_{{R_j} \in pa\left( {{R_i}} \right)} {{v_{ij}}} P\left( {R_j^\prime \mid Q} \right)

(3) Calculate the correlation P(B_j|Q) between query information Q and document B_j, i.e., the final a posteriori probability of document B_j, through equation (17): 17P(Bj∣Q)=∑Ri∈pa(Bj)wijP(ri∣Q)P\left( {{B_j}\mid Q} \right) = \sum\limits_{{R_i} \in pa\left( {{B_j}} \right)} {{w_{ij}}} P\left( {{r_i}\mid Q} \right)

Acquired and query information Q the highest degree of relevance of the document B_j that is the most relevant information with the user query document, that is, the user needs the document, through the above process to realize the library information retrieval.

A total of 862 digital resources from the digital library of University E were randomly selected as the experimental document data, with a total of 2015 terms appearing in their titles, 597 terms with document frequency >2, and 643 pairs of related terms with a term number of 2 (the number of documents with related terms appearing in their titles >2). There were 445 pairs of related terms with a correlation of less than 1 calculated using the assumption of interterm independence, where the correlation is the number of documents in which both terms appear in the title. Most of the related terms were proper nouns.

When the user query is 5 terms and the terms are all frequent itemsets, a total of 306 frequent itemsets are calculated. Using (S3) the retrieval system proposed in this paper calculates that there is an average of 5.91 number of useful documents, an average of 5.68 number of actual useful documents, and (S4) an average of 2.128 number of documents under the assumption that the terms are independent of each other. Arbitrarily selected 15 of the same type of query calculation results are shown in Table 1, overall (S3) the proposed retrieval system calculates the number of documents closer to the actual number of useful documents, the difference is controlled in [0,3].

Set the critical value of similarity between user query and document as 0.7, select 12 identical queries and compare the engine usefulness computed by (S1) the systematic approach of this paper and (S4) the engine usefulness computed under the assumption of mutual independence between terms are shown in Table 2. (S3) The retrieval system proposed in this paper not only calculates the same number of documents as the actual number of useful documents, but also maintains the average similarity at 0.70 and above, up to 0.86, which shows greater feasibility in vector queries.

To further test the response performance of the proposed retrieval system, 20 retrieval experiments are conducted using (S3) the proposed retrieval system in this paper, and its response time performance is shown in Fig. 6. In the 20 retrieval experiments, the retrieval system in this paper has certain fluctuations, but the overall response time is more ideal, controlled in the [0,1.25] ms interval, and in the 13th experiment is as low as 0.24 ms, which has a high retrieval efficiency, and shows excellent applicability in practical scenarios.

Figure 6.

Operational efficiency assessment

The overall Chinese literature resources of the digital library of university E were used as the test data, which contained a total of 350,000 literature data in economy, agriculture, industry, and medicine, according to which eight search terms were selected: (Q1) trade, (Q2) finance, (Q3) edible fungus, (Q4) wheat, (Q5) electric power, (Q6) communication, (Q7) biomaterials, and (Q8) ultrasound, and were still used as the control by (S1) a multi Agent technology-based library retrieval system, (S2) cognitive style-based library retrieval system as a control. The accuracy of the search results of the three retrieval systems in the digital library of university E is shown in Fig. 7.

Figure 7.

The retrieval accuracy of the three retrieval systems

The accuracy of the retrieval results of (S3) the retrieval system proposed in this paper was consistently 96.00% and above on the search terms of eight different domains, and the retrieval accuracy of (S2) the cognitive style-based library retrieval system was slightly inferior to that of (S3) the retrieval system proposed in this paper, with the highest retrieval results of 95.00%. In contrast, (S1) Multi-Agent technology based library retrieval system has the worst retrieval accuracy (<90.00%). (S3) The retrieval system proposed in this paper not only has the highest retrieval accuracy, but also floats less, which is the best among the three retrieval systems. Thanks to the incorporation of quantum Bayesian network, (S3) the retrieval system proposed in this paper has lower computational complexity and thus higher retrieval accuracy on rich types of resources.

Using the same search term, (S1) a library retrieval system based on multi-agent technology, (S2) a library retrieval system based on cognitive style. Setting up 20 experiments, the retrieval time is incremented in frequency of 3 s. The retrieval time and the number of retrieved documents for the three retrieval systems are shown in Fig. 8. When the retrieval time is the same, the number of documents retrieved by (S3) the retrieval system proposed in this paper (> 140) is always significantly more than that by the other retrieval models (80~ 120). When the retrieval time reaches 51s, the results of all retrieval systems tend to stabilize, and the number of retrieved documents of (S3) the proposed retrieval system is also stabilized at about 175, which is far beyond the other two retrieval systems.

Figure 8.

The retrieval time and quantity of the three models