CN113553027A - Random number generation method based on real-time estimation of stacking state preparation error rate in quantum computer - Google Patents

Random number generation method based on real-time estimation of stacking state preparation error rate in quantum computer Download PDF

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CN113553027A
CN113553027A CN202110673430.XA CN202110673430A CN113553027A CN 113553027 A CN113553027 A CN 113553027A CN 202110673430 A CN202110673430 A CN 202110673430A CN 113553027 A CN113553027 A CN 113553027A
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李元昊
费洋扬
王卫龙
王洪
马智
孟祥栋
段乾恒
王娜
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Abstract

The invention relates to the technical field of quantum information communication, and discloses a random number generation method based on real-time estimation of a superposition state preparation error rate in a quantum computer, which comprises the following steps: preparing the qubit into a superposition state using a RY (pi/2) gate in an initial state |0>
Figure DDA0003119720710000011
And transmitting the prepared superposition state to a credible measuring terminal; randomly selecting an X base or a Z base to measure the superposition state by using a string of random seeds; estimating the preparation error rate e of the superposition state in the X-base measuring result in real time according to the X-base measuring resultbxAnd according to ebxError rate e for stacked state preparation under Z basezCarrying out estimation; generating a random number by using a measurement result under the Z base; and (4) performing randomness extraction on the generated original data by using a Toeplitz post-processing method. The invention relates to aThe error rate of the prepared superposition state is effectively estimated in real time, so that the minimum entropy in the quantum computer is monitored in real time, and the randomness of the generated random number is ensured.

Description

Random number generation method based on real-time estimation of stacking state preparation error rate in quantum computer
Technical Field
The invention relates to the technical field of quantum information communication, in particular to a random number generation method based on real-time estimation of a superposition state preparation error rate in a quantum computer.
Background
Random numbers are widely used in modern society, and play an important role in many fields, such as monte carlo analog sampling, gaming, numerical computation simulation, information security, etc., and especially in Cryptography and secure communications (refer j. menezes, score a. vanstone, Paul c. van oorschot. Handbook of Applied cryptograph [ M ]// Handbook of Applied cryptograph, 1997.) (schinier b. Applied cryptograph [ M ]. Wiley John + Sons, 2009).
In the prior art, random numbers generated by means of classical deterministic mathematical algorithms or according to classical physical processes are called pseudo-random numbers. Although pseudo-random numbers can satisfy the statistical properties of random numbers, the generation principle is a deterministic process, and the generated random numbers are actually predictable and not true random numbers. Therefore, in a field where the requirement for randomness of random numbers is relatively high, the security of pseudo-random numbers cannot meet the requirements of these applications.
Quantum random numbers generated according to the quantum mechanics principle have also been proposed in the prior art, and are considered as true random numbers whose information theory is theoretically secure and provable. The existing quantum random number generation protocol is mainly realized based on quantum optical devices, for example, schemes such as single photon detection, vacuum fluctuation and laser phase noise are utilized. On the other hand, research on physical implementation of quantum computers, especially quantum computer schemes based on superconducting circuit implementation, is rapidly advancing. Meanwhile, quantum cloud computing of hewlett packard services such as quantum computing hardware and software provides convenience for achieving quantum computing capacity. Quantum stacking states and quantum entanglement resources present in quantum computers make it possible for quantum computers to generate quantum random numbers.
However, current quantum computers are noisy, with errors in initial state preparation, quantum gate operation, and quantum readout operations. In the existing random number generation protocol based on a superconducting quantum computer, an initial state is generally prepared into a superposition state through a Hadamard quantum gate, the superposition state is repeatedly measured to obtain a random number, the randomness of the generated random number depends on the performance of the quantum computer, and the prepared superposition state cannot be estimated. The existing protocol for generating random number by using quantum computer is characterized by that firstly, preparing initial state, utilizing Hadamard quantum logic gate operation to prepare initial state into superposition state
Figure BDA0003119720690000021
In calculating the base |0>And |1>And measuring the superposition state to obtain 0 or 1, and forming a random number by the results after multiple measurements. However, the existing quantum computer produces the superposition state | +due to environmental noise or imperfect control mechanism>There is a certain error, that is, the prepared quantum state is measured for many times, so that a sequence with balanced 1 and 0 cannot be obtained, and the safety and randomness of the formed random number cannot be ensured.
Disclosure of Invention
The invention provides a random number generation method based on real-time estimation of a preparation error rate of a superposition state in a quantum computer, aiming at the problems that in the conventional random number generation protocol based on a superconducting quantum computer, an initial state is generally prepared into the superposition state through a Hadamard quantum gate, the superposition state is repeatedly measured to obtain a random number, the randomness of the generated random number depends on the performance of the quantum computer, and the prepared superposition state cannot be estimated.
In order to achieve the purpose, the invention adopts the following technical scheme:
a random number generation method based on a superposition state preparation error rate in a real-time estimation quantum computer comprises the following steps:
step 1: in the initial state |0>Preparing quantum bit to superposition state by applying RY (pi/2) gate
Figure BDA0003119720690000022
And transmitting the prepared superposition state to a credible measuring terminal;
step 2: the superposition state is measured by randomly selecting X base or Z base by using a string of random seeds, and the quantum line runs n times in total, including nxQuantum wires for sub-X-base measurements and nzA quantum wire for sub-Z-basis measurement;
and step 3: estimating the preparation error rate e of the superposition state in the X-base measuring result in real time according to the X-base measuring resultbxAnd according to ebxError rate e for stacked state preparation under Z basezCarrying out estimation;
and 4, step 4: generating a random number using the measurement result under the Z basis to form a random sequence in which a number 0 indicates that the measurement result is |0>The number 1 indicates that the measurement result is |1>In which the quantum wires of the Z-based measurement run in total nzThen, n can be generatedzA random bit;
and 5: and (4) performing randomness extraction on the generated original data by using a Toeplitz post-processing method.
Further, the method estimates the preparation error rate e of the superposition state in the X-base measurement result in real time according to the X-base measurement resultbxThe method comprises the following steps:
the number of |0> and |1> without read errors is calculated from the known X-base measurements:
Figure BDA0003119720690000031
wherein N is0And N1Respectively represent |0 in the X-base measurement results>And |1>And satisfies N0+N1＝nx；n0And n1Denotes |0 in the X-base measurement in the ideal case without readout errors>And |1>The number of (2); r is0Represents |0>Read error rate of (2), i.e. prepared is |0>But the result of the readout is |1>The proportion of the components is calculated; r is1Represents |1>Read error rate of (2), i.e. prepared is |1>But the result of the readout is |0>The proportion of the components is calculated;
n0and n1Satisfies the following formula:
Figure BDA0003119720690000032
the error rate of the stack preparation in the X-based measurement result, namely the probability of appearance of | - > is as follows:
Figure BDA0003119720690000033
Figure BDA0003119720690000034
where δ represents the angle of rotation about the Y axis
Figure BDA0003119720690000035
The angle is offset and phi represents the offset angle of rotation about the Z axis.
Further, said is according to ebxError rate e for stacked state preparation under Z basezThe estimating includes:
error rate e of stacked state preparation under Z basezSatisfies the following formula:
ez≤ebx+θ
wherein θ represents an error caused by statistical fluctuation, and satisfies the following formula:
Figure BDA0003119720690000036
wherein ζ (θ) ═ H (e)bx+θ+qxθ)-qxH(ebx)-(1-qx)H(ebx+θ)，qx＝nxThe/n is the ratio of the selected X-based measurement, and H () represents Shannon entropy; epsiloneIs a given fixed parameter indicating the probability of failure; prob (e)z>ebx+ theta) represents the error rate e of the preparation of the superposed state in the Z radicalzGreater than ebxProbability of + θ, i.e. pair ezThe value of (c) estimates the probability of failure.
Compared with the prior art, the invention has the following beneficial effects:
in the prior art, various noises existing in a quantum computer have a large influence on the randomness of the generated random numbers, the relationship between the two has no clear quantitative description, and the randomness of the generated random numbers may become poor under the condition of large noises. In the present invention, random numbers with provable randomness can be generated even if various noises exist in a quantum computer, and the control of quantum gates and the output operation of quantum states are imperfect. By effectively estimating the error rate of the prepared superposition state in real time, the minimum entropy in the quantum computer is monitored in real time, and the randomness of the generated random number is ensured.
Drawings
FIG. 1 is a basic flowchart of a random number generation method based on real-time estimation of error rate of stack state preparation in a quantum computer according to an embodiment of the present invention;
FIG. 2 is a structural topology diagram of qubits of a quantum computer used in experiments;
FIG. 3 is a graph of φ as a function of δ;
FIG. 4 is ebxGraph as a function of δ.
Detailed Description
The invention is further illustrated by the following examples in conjunction with the accompanying drawings:
as shown in fig. 1, a random number generation method based on real-time estimation of error rate of superposition state preparation in a quantum computer includes:
step S101, a random source: in the initial state |0>Preparing quantum bit to superposition state by applying RY (pi/2) gate
Figure BDA0003119720690000041
And transmitting the prepared superposition state to a credible measuring terminal;
in particular, due to imperfections in the physical implementation of quantum computers, the single quantum bit gate RY (pi/2) used is in error. The error can be divided into two parts, one part is the angle of rotation around the Y axis and
Figure BDA0003119720690000042
the angle has a deviation delta and the other part is the deviation from the Y axis of the rotation axis, corresponding to the deviation angle phi of the rotation around the Z axis. Thus, an actual RY (π/2) gate may be equivalent to
Figure BDA0003119720690000051
The superposition of gates and RZ (phi) gates, expressed in matrix form as
Figure BDA0003119720690000052
And
Figure BDA0003119720690000053
errors in the operation of the RY (pi/2) gate can cause errors in the prepared stack state, namely the parameter ezThe value of (c) needs to be estimated. In the random source part, the qubit is first prepared to the ground state |0>After passing through the imperfect RY (pi/2) gate, the qubit is in the state
Figure BDA0003119720690000054
Thus, the quantum state emitted from the random source is at | +>Sum of states | ->Superposition of states, where | +>The probability of a state is
Figure BDA0003119720690000055
|->The probability of a state is
Figure BDA0003119720690000056
Step S102, random sampling: the superposition state is measured by randomly selecting X base or Z base by using a string of random seeds, and the quantum line runs n times in total, including nxQuantum wires for sub-X-base measurements and nzA quantum wire for sub-Z-basis measurement; it is worth noting that in quantum computers, by adding a RY (pi/2) gate, the measurement under the Z-basis can be converted into an X-basis measurement, where
Figure BDA0003119720690000057
Z＝{|0>,|1>}。
In particular, in quantum wires for X-based measurements, the same imperfect RY (π/2) gate is again used to act on the quantum state
Figure BDA0003119720690000058
At this point the quantum state becomes:
Figure BDA0003119720690000059
for quantum state
Figure BDA00031197206900000510
Measurement is performed to obtain |0>Or |1>. Theoretically, |0 is obtained>Has a probability of
Figure BDA0003119720690000061
To obtain |1>Has a probability of
Figure BDA0003119720690000062
Step S103, parameter estimation: estimating the preparation error rate e of the superposition state in the X-base measuring result in real time according to the X-base measuring resultbxAnd according to ebxFor superimposed state system under Z baseSpare error rate ezCarrying out estimation;
specifically, in the case of a quantum computer without noise, the quantum states emitted by the random source should all be in the superposition state | + >. Under X, the measurement of | plus > is |1> and the measurement of | minus > is |0 >. Therefore, if the result of X-base measurement is |0>, it represents that there is an error in the preparation of the superposed state.
In quantum computers, readout errors for quantum states are also not negligible. Remember |0>Has a read error rate of r0Denotes that prepared is |0>But the result of the readout is |1>In a similar manner, let us say |1>Has a read error rate of r1Prepared is |1>But the result of the readout is |0>The ratio of the active ingredients to the total amount of the active ingredients. From the measurement of the X base, the following equation can be obtained:
Figure BDA0003119720690000063
wherein N is0And N1Respectively represent |0 in the X-base measurement results>And |1>And satisfies N0+N1＝nx；n0And n1Denotes |0 in the X-base measurement in the ideal case without readout errors>And |1>The number of (2); n is0And n1Satisfies the following formula:
Figure BDA0003119720690000064
according to quantum state
Figure BDA0003119720690000065
In the form of | ->The probability of state occurrence, so that the error rate of the preparation of the superposition state in the X-base measurement result, namely | ->The probability of occurrence is:
Figure BDA0003119720690000066
it is worth noting that |0 in the X-base measurement results>And |1>Number N of0And N1It is known that n can be calculated from equation (3)0And n1The value of (c). In a quantum computer, the value ranges of delta and phi are
Figure BDA0003119720690000071
So that 0 can be obtained<cos(φ)<1; further, according to the formula (4), a
Figure BDA0003119720690000072
Therefore, δ satisfies the relation:
Figure BDA0003119720690000073
wherein the value range of delta can be determined. The value of δ, φ given a certain δ, can also be calculated according to equation (4). Thus, according to expression (5), e can be estimatedbxAnd gives the maximum value.
And then the error rate e can be prepared according to the superposition state in the X-base measurement resultbxError rate e for preparation of a superposed State in the Z basezThe estimation is performed, and the formula (7) is satisfied:
ez≤ebx+θ (7)
wherein θ represents an error caused by statistical fluctuation, and satisfies formula (8):
Figure BDA0003119720690000074
wherein ζ (θ) ═ H (e)bx+θ+qxθ)-qxH(ebx)-(1-qx)H(ebx+θ)，qx＝nxThe/n is the ratio of the selected X-based measurement, and H () represents Shannon entropy; epsiloneIs a given fixed parameter indicating the probability of failure; prob (e)z>ebx+ theta) represents the error rate e of the preparation of the superposed state in the Z radicalzGreater than ebxProbability of + θ, i.e. pair ezThe value of (c) estimates the probability of failure.
Step S104, generating randomness: generating a random number using the measurement result under the Z basis to form a random sequence in which a number 0 indicates that the measurement result is |0>The number 1 indicates that the measurement result is |1>In which the quantum wires of the Z-based measurement run in total nzThen, n can be generatedzA random bit;
step S105, randomness extraction: and (4) performing randomness extraction on the generated original data by using a Toeplitz post-processing method. The number of random bits K that can be extracted at the end is:
K＝nz-nzH(ez)-te (9)
wherein t iseIs the probability of a randomness extraction failure.
To verify the effect of the present invention, the following experiment was performed:
according to the method of the invention, an experiment is carried out using the quantum computer cloud platform of IBM. To demonstrate the effectiveness of the present invention, we performed 251X-base measurements directly after 8192Z-base measurements on qubits. The quantum computer used in the experiment is IBMQ _5_ yorktown, the structural topological diagram of the quantum bit of the IBMQ _5_ yorktown quantum device is shown in figure 2, and the device has 5 quantum bits Q0、Q1、Q2、Q3、Q4Wherein, 0, 1, 2, 3, 4 respectively represent the corresponding quantum bit Q0、Q1、Q2、Q3、Q4And 0 represents a qubit Q0In qubit Q0Run a quantum program.
By repeatedly measuring and operating the quantum circuit, two groups of sequences are finally obtained, namely the measurement result l of the Z basezLength 819200; and measurement result l of X radicalxAnd the length is 25100. In the sequence lxNumber of (1) to (0) N0Is a number N equal to 2669, 11Equal to 22431. In qubit Q0For quantum state |0>Read error rate r of0Is 0.072, for quantum state |1>Read error rate r of1Is 0.0394. According to formula (3)
Figure BDA0003119720690000081
The number n of 0's in the X-based measurement result can be calculated without readout error0Number n of 299.8557, 1124800.1443. Then according to formula (6)
Figure BDA0003119720690000082
The deviation angle δ, which can be calculated about the Y-axis, is in the range-0.1095186<δ<0.1095186. The value of phi can also be calculated for each given determined delta, phi, according to equation (4), as shown in fig. 3. Using error rate ebxThe corresponding error rate e of the preparation of the stack state can be calculated by the expression (5)bxE can be calculated as shown in FIG. 4bxIs 0.011943.
In conclusion, the invention realizes effective estimation of the error rate of the superposition state prepared in quantum computation. And measuring the prepared superposition state | + > under an X base or a Z base by using a random selection method, wherein the measurement result of the X base is used for estimating the error rate of the prepared superposition state, and the measurement result of the Z base is used for generating a random number. With the known number of 0's and 1's in the X radical, the error rate of the prepared stacked state can be accurately estimated and its upper bound given.
In the prior art, various noises existing in a quantum computer have a large influence on the randomness of the generated random numbers, the relationship between the two has no clear quantitative description, and the randomness of the generated random numbers may become poor under the condition of large noises. In the present invention, random numbers with provable randomness can be generated even if various noises exist in a quantum computer, and the control of quantum gates and the output operation of quantum states are imperfect. By effectively estimating the error rate of the prepared superposition state in real time, the minimum entropy in the quantum computer is monitored in real time, and the randomness of the generated random number is ensured.
The above shows only the preferred embodiments of the present invention, and it should be noted that it is obvious to those skilled in the art that various modifications and improvements can be made without departing from the principle of the present invention, and these modifications and improvements should also be considered as the protection scope of the present invention.

Claims (3)

1. A random number generation method based on a superposition state preparation error rate in a real-time estimation quantum computer is characterized by comprising the following steps:
step 1: in the initial state |0>Preparing quantum bit to superposition state by applying RY (pi/2) gate
Figure FDA0003119720680000011
And transmitting the prepared superposition state to a credible measuring terminal;
step 2: the superposition state is measured by randomly selecting X base or Z base by using a string of random seeds, and the quantum line runs n times in total, including nxQuantum wires for sub-X-base measurements and nzA quantum wire for sub-Z-basis measurement;
and step 3: estimating the preparation error rate e of the superposition state in the X-base measuring result in real time according to the X-base measuring resultbxAnd according to ebxError rate e for stacked state preparation under Z basezCarrying out estimation;
and 4, step 4: generating a random number using the measurement result under the Z basis to form a random sequence in which a number 0 indicates that the measurement result is |0>The number 1 indicates that the measurement result is |1>In which the quantum wires of the Z-based measurement run in total nzThen, n can be generatedzA random bit;
and 5: and (4) performing randomness extraction on the generated original data by using a Toeplitz post-processing method.
2. The method as claimed in claim 1, wherein the method for generating random number based on real-time estimation of error rate of preparation of superposition state in quantum computer is characterized in that the real-time estimation of error rate e of preparation of superposition state in X-base measurement resultbxThe method comprises the following steps:
the number of |0> and |1> without read errors is calculated from the known X-base measurements:
Figure FDA0003119720680000012
wherein N is0And N1Respectively represent |0 in the X-base measurement results>And |1>And satisfies N0+N1＝nx；n0And n1Denotes |0 in the X-base measurement in the ideal case without readout errors>And |1>The number of (2); r is0Represents |0>Read error rate of (2), i.e. prepared is |0>But the result of the readout is |1>The proportion of the components is calculated; r is1Represents |1>Read error rate of (2), i.e. prepared is |1>But the result of the readout is |0>The proportion of the components is calculated;
n0and n1Satisfies the following formula:
Figure FDA0003119720680000021
the error rate of the stack preparation in the X-based measurement result, namely the probability of appearance of | - > is as follows:
Figure FDA0003119720680000022
Figure FDA0003119720680000023
where δ represents the angle of rotation about the Y axis
Figure FDA0003119720680000024
The angle is offset and phi represents the offset angle of rotation about the Z axis.
3. The method of claim 1, wherein the error rate is estimated based on the randomness of the stacking state preparation in the quantum computerNumber generation method, characterized in that said basis is ebxError rate e for stacked state preparation under Z basezThe estimating includes:
error rate e of stacked state preparation under Z basezSatisfies the following formula:
ez≤ebx+θ
wherein θ represents an error caused by statistical fluctuation, and satisfies the following formula:
Figure FDA0003119720680000025
wherein ζ (θ) ═ H (e)bx+θ+qxθ)-qxH(ebx)-(1-qx)H(ebx+θ)，qx＝nxThe/n is the ratio of the selected X-based measurement, and H () represents Shannon entropy; epsiloneIs a given fixed parameter indicating the probability of failure; prob (e)z＞ebx+ theta) represents the error rate e of the preparation of the superposed state in the Z radicalzGreater than ebxProbability of + θ, i.e. pair ezThe value of (c) estimates the probability of failure.
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