A low failure rate quantum algorithm for searching maximum or minimum

Abstract

Although Durr and Hoyer have proposed state-of-the-art quantum algorithm (DHA) for searching minimum value, the lower limit of DHA’s successful probability is 1/2 . Also, DHA requires approximately $(\log _{2}N)^2$ copies of the initial state. In this paper, we propose a new quantum maximum or minimum searching algorithm (QUMMSA). In big data scenarios, according to sparse sampling with different densities, we can estimate the corresponding precision parameters. QUMMSA can improve the successful probability close to $100\%$. Furthermore, with the quantum exact search algorithm, QUMMSA only requires approximately $\log _2 N$ copies of the initial state to solve this problem. Since preparing an arbitrary quantum state is a problem with exponential complexity, our algorithm has a greater advantage with the increasing database size. In addition, we first propose a general method for circuits construction, which can be used in any database. An experiment implemented in an IBM superconducting processor and a numerical simulation of a 6-qubit system to solve a real issue indicate the feasibility and efficiency of QUMMSA. QUMMSA can serve as a subroutine in various quantum algorithms which involves searching maximum or minimum.

Appendices

Appendix A: Review of Grover–Long algorithm

In this section, we have to review Grover–Long algorithm which is a key step of QUMMSA, so that we can understand how to calculate the parameters of Grover–Long algorithm and apply it to QUMMSA.

Firstly, the initial state can be prepared by W operator, which can be described as formula(A1):

$$\begin{aligned} \left| \Psi \right\rangle =W\left| 0^{\otimes n}\right\rangle =\frac{1}{\sqrt{N}}\sum _{i=0}^{N-1}\left| i\right\rangle =\sqrt{\frac{M}{N}}\left| \Psi _{\mathrm{good}}\right\rangle +\sqrt{\frac{N-M}{N}}\left| \Psi _{\mathrm{bad}}\right\rangle \end{aligned}$$

(19)

where $\left| \Psi _{\mathrm{good}}\right\rangle$ stores solutions which we want to find and $\left| \Psi _{\mathrm{bad}}\right\rangle$ stores other values. Specifically, if we want to find the minimum, $\left| \Psi _{\mathrm{good}}\right\rangle$ stores all values of database that are less than or equal to $d_0$; $\left| \Psi _{\mathrm{bad}}\right\rangle$ stores all values of database that are greater than $d_0$, in contrast. Especially, when $N=2^n$, the initial state is a uniform superposition state, the W operator becomes $H^{\otimes n}$, where H is the Walsh–Hadamard transformation; n is the number of qubits. In this special case, the complexity of preparing the initial state is $\log _2 N$ (i.e., the initial state can be prepared by $\log _2 N$ H operators).

One Grover iteration G can be divided into four operators  [25].

$$\begin{aligned} G = -WI_0W^{-1}O \end{aligned}$$

(20)

where O is an oracle which performs a phase inversion on $\left| \Psi _{\mathrm{good}}\right\rangle$; $I_0$ is a conditional phase shift operator which performs a phase inversion on $\left| 0\right\rangle$.

Grover–Long algorithm is done by replacing the phase inversion with an adjustable angle $\phi$ phase rotation. The rotation angle is given as:

$$\begin{aligned} \phi =2 \mathrm{arcsin}\left( \frac{\sin \frac{\pi }{4J+2} }{\sin \beta } \right) \end{aligned}$$

(21)

where $\sin \beta =\sqrt{\frac{M}{N}}$. Upon measurement in Jth iteration, one of marked states is obtained with zero failure rate.

$$\begin{aligned} J\ge \mathrm{floor}\left( \frac{\frac{\pi }{2}-\beta }{\beta }\right) +1 \end{aligned}$$

(22)

By utilizing the number of solutions M and the database size N, we can calculate the exact value of $\beta$, $\phi$, J. Grover–Long algorithm will find a solution with zero failure rate.

Appendix B: Theoretical analysis for the failure rate

In this section, we given the theoretical failure rate of Grover–Long algorithm and QESA, when M/N is unknown.

Firstly, we present a proof of Grover–Long algorithm’s failure rate $\varepsilon _{\mathrm{GL}}$, when M is unknown. The performance is shown in Fig. 6. The initial quantum state is expressed as:

$$\begin{aligned} \left| \Psi \right\rangle = \left( \sqrt{ \frac{M}{N} } \left| \Psi _{\mathrm{good}}\right\rangle + \sqrt{ \frac{N-M}{N} } \left| \Psi _{\mathrm{bad}}\right\rangle \right) \end{aligned}$$

(23)

where $\left| \Psi _{\mathrm{good}}\right\rangle$ includes M solutions. $\left| \Psi _{\mathrm{bad}}\right\rangle$ includes $N-M$ non-solutions. Each quantum state store a data value and their amplitude is expressed as $\alpha _{\mathrm{good}}^{(0)}=\alpha _{\mathrm{bad}}^{(0)}=1/\sqrt{N}$, in the initial state. Otherwise, the amplitude will be 0, if no data value is stored.

Grover–Long algorithm is divided into 4 steps. The first step is the oracle operator. It makes solutions receive a phase shift $\phi$:

$$\begin{aligned} O \left| \Psi \right\rangle = \left[ \begin{array}{c} {\mathrm{e}}^{i\phi } \quad \quad \\ \qquad 1 \end{array} \right] \left( \left| \Psi _{\mathrm{good}}\right\rangle + \left| \Psi _{\mathrm{bad}}\right\rangle \right) \end{aligned}$$

(24)

The steps (2), (3), (4) can be expressed as:

$$\begin{aligned} -W\left[ \left( {\mathrm{e}}^{i\phi }-1 \right) \left| 0\right\rangle \left\langle 0\right| \right] W^{-1} =\left( {\mathrm{e}}^{i\phi }-1 \right) \left| \Psi \right\rangle \left\langle \Psi \right| -I \end{aligned}$$

(25)

Therefore, Grover–Long algorithm can be expressed as:

$$\begin{aligned} G\left| \Psi \right\rangle \left\langle \Psi \right| = \left[ -\left( {\mathrm{e}}^{i\phi }-1 \right) \left| \Psi \right\rangle \left\langle \Psi \right| -I \right] \left( {\mathrm{e}}^{i\phi } \left| \Psi _{\mathrm{good}}\right\rangle +\left| \Psi _{\mathrm{bad}}\right\rangle \right) \end{aligned}$$

(26)

Because M/N is unknown in QUMMSA, we use the estimated value $\widetilde{M} / \widetilde{N}$ to calculate the estimated parameters $\widetilde{\beta }$, $\widetilde{\phi }$, $\widetilde{J}$ of Grover–Long algorithm. The gap between the exact parameters and the estimated parameters will lead the failure rate of Grover–Long algorithm. Meanwhile, it is noteworthy that all quantum states as less than or equal to $d_0$ are marked, regardless of whether the quantum state stores a data value. Thus, $\widetilde{M} \ge M$. Marking quantum states of 0 amplitude does not affect the iterative process. If the amplitude of a quantum state is 0, the amplitude will be still 0 after the amplitude amplification. Therefore, even if $\widetilde{M} \ne M$, the oracle can still correctly mark the values in the database that are less than or equal to $d_0$.

After G operator, we can get two results. For solutions, each quantum state’s amplitude $\alpha _{\mathrm{good}}^{(j)}$ will be expressed as:

$$\begin{aligned} \alpha _{\mathrm{good}}^{(j)}= & {} -\alpha _{\mathrm{good}}^{(j-1)} \times \left( 1+ \frac{{\mathrm{e}}^{i \widetilde{\phi }}-1}{N} \right) - \alpha _{\mathrm{good}}^{(j-1)} (M-1) \times \frac{{\mathrm{e}}^{i \widetilde{\phi }}-1}{N} - \alpha _{\mathrm{bad}}^{(j-1)} \nonumber \\&\times (N-M) \times \frac{{\mathrm{e}}^{i \widetilde{\phi }}-1}{N} \end{aligned}$$

(27)

For non-solutions, each quantum state’s amplitude $\alpha _{\mathrm{bad}}^{(j)}$ will be expressed as:

$$\begin{aligned} \alpha _{\mathrm{bad}}^{(j)}= & {} -\alpha _{\mathrm{bad}}^{(j-1)} \times \left( 1+ \frac{{\mathrm{e}}^{i \widetilde{\phi }}-1}{N} \right) - \alpha _{\mathrm{good}}^{(j-1)} \times M \times \frac{{\mathrm{e}}^{i \widetilde{\phi }}-1}{N} - \alpha _{\mathrm{bad}}^{(j-1)} \nonumber \\&\times (N-M-1) \times \frac{{\mathrm{e}}^{i \widetilde{\phi }}-1}{N} \end{aligned}$$

(28)

where j is the current number of Grover–Long iteration and $j \in [1,\widetilde{J}]$.

The failure rate is expressed as:

$$\begin{aligned} \varepsilon _{\mathrm{GL}} = 1- \left| \alpha _{\mathrm{good}}^{\widetilde{J}} \right| ^{2} \times M \end{aligned}$$

(29)

Secondly, we present proof of QESA’s failure rate $\varepsilon _{\mathrm{ESA}}$. Due to the unknown M, different number of Grover iterations is selected in different possibility in once QESA iteration. Specially, Ref  [15] set a parameter $\lambda \in (1,4/3]$. The number of Grover iterations v is a random number which is selected from $[0,\lambda ^{t-1}]$ and rounds down where t is the current number of QESA iteration.

In the first QESA iteration.

$$\begin{aligned} \varepsilon _{\mathrm{ESA}}^{(1)}=1- \frac{M}{N} \end{aligned}$$

(30)

If the algorithm does not find a correct solution, it will run forever. Meanwhile, $\varepsilon _{\mathrm{ESA}}^{(t)}$ is decreased with the increase of t.

$$\begin{aligned} \varepsilon _{\mathrm{ESA}}^{(t)}= & {} \varepsilon _{\mathrm{ESA}}^{(t-1)} \times \left\{ \lambda ^{-t+1} \left( 1-\frac{M}{N} \right) + \sum _{v=1}^{\mathrm{floor}\left( \lambda ^{t-1} \right) -1} \left[ \lambda ^{-t+1} \cos ^2 \left( (2v+1) \times \arcsin \sqrt{ \frac{M}{N}} \right) \right] \right. \nonumber \\&+\left. \left[ \lambda ^{t-1} -\mathrm{floor}\left( \lambda ^{t-1} \right) \right] \times \cos ^2 \left[ \left( 2 \times \mathrm{floor} \left( \lambda ^{t-1} \right) + 1 \right) \times \arcsin \sqrt{\frac{M}{N}} \right] \right\} \end{aligned}$$

(31)

If $\lambda ^{t-1}>\sqrt{N}$, then $\sqrt{N}$ will replace $\lambda ^{t-1}$.

Proving the failure rates of the two algorithm not only provides theoretical support for our experiments, but also we can obtain the theoretical failure rate of the two algorithms, when they are applied to any database.

Appendix C: The parameters of IBM quantum superconducting processor

In this section, we present some parameters of IBM quantum superconducting processor. The schematic and topology of this processor are shown in Fig. 12a, b, respectively. Two co-planar waveguide (CPW) resonators, acting as quantum buses, provide the device control and readout. Entanglement in IBM system is achieved via CNOT gates, which use cross-resonance  [38, 39]. Single qubit rotation gate with an arbitrary angle and CNOT are as primitive operators. Single qubit gate and multi qubit gate error of different qubits are shown in Table 3.

Fig. 12

5-qubit superconducting processor: a schematic; b topology

Table 3 The detail parameters of IBM quantum superconducting processor

Appendix D: Data sources

This section shows data source of the second demo. Complete data can be obtained on Kaggle website (https://www.kaggle.com/c/titanic/data). The complete data we use are listed in Table 4.

Table 4 Titanic passengers age (excerpt)