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Cross-entropy benchmarking (also referred to as XEB) is a quantum benchmarking protocol which can be used to demonstrate quantum supremacy.^[1] In XEB, a random quantum circuit is executed on a quantum computer multiple times in order to collect a set of $k$ samples in the form of bitstrings $\{x_{1},\dots ,x_{k}\}$ . The bitstrings are then used to calculate the cross-entropy benchmark fidelity ( $F_{\rm {XEB}}$ ) via a classical computer, given by

F_{\rm {XEB}}=2^{n}\langle P(x_{i})\rangle _{k}-1={\frac {2^{n}}{k}}\left(\sum _{i=1}^{k}|\langle 0^{n}|C|x_{i}\rangle |^{2}\right)-1

,

where $n$ is the number of qubits in the circuit and $P(x_{i})$ is the probability of a bitstring ${x_{i}}$ for an ideal quantum circuit $C$ . If $F_{XEB}=1$ , the samples were collected from a noiseless quantum computer. If $F_{\rm {XEB}}=0$ , then the samples could have been obtained via random guessing.^[2] This means that if a quantum computer did generate those samples, then the quantum computer is too noisy and thus has no chance of performing beyond-classical computations. Since it takes an exponential amount of resources to classically simulate a quantum circuit, there comes a point when the biggest supercomputer that runs the best classical algorithm for simulating quantum circuits can't compute the XEB. Crossing this point is known as achieving quantum supremacy; and after entering the quantum supremacy regime, XEB can only be estimated.^[3]

The Sycamore processor was the first to demonstrate quantum supremacy via XEB. Instances of random circuits with $n=53$ and 20 cycles were run to obtain an XEB of $0.0024$ .^[3] Generating samples took 200 seconds on the quantum processor when it would have taken 10,000 years on Summit at the time of the experiment. Improvements in classical algorithms have shortened the runtime to about a week on Sunway TaihuLight thus collapsing Sycamore's claim to quantum supremacy.^[4] As of 2021, the latest demonstration of quantum supremacy by Zuchongzhi 2.1 with $n=60$ , 24 cycles and an XEB of $0.000366$ holds. It takes around 4 hours to generate samples on Zuchongzhi 2.1 when it would take 10,000 years on Sunway.^[4]

References

^ Boixo, S.; et al. (2018). "Characterizing Quantum Supremacy in Near-Term Devices". Nature Physics. 14 (6): 595–600. arXiv:1608.00263. Bibcode:2018NatPh..14..595B. doi:10.1038/s41567-018-0124-x. S2CID 4167494.
^ Aaronson, S. (2021). "Open Problems Related to Quantum Query Complexity". arXiv:2109.06917 [quant-ph].
^ ^a ^b Arute, F.; et al. (2019). "Quantum supremacy using a programmable superconducting processor". Nature. 574 (7779): 505–510. arXiv:1910.11333. Bibcode:2019Natur.574..505A. doi:10.1038/s41586-019-1666-5. PMID 31645734. S2CID 204836822.
^ ^a ^b Liu, X.; et al. (2021). "Redefining the Quantum Supremacy Baseline With a New Generation Sunway Supercomputer". arXiv:2111.01066 [quant-ph].

[1] Boixo, S.; et al. (2018). "Characterizing Quantum Supremacy in Near-Term Devices". Nature Physics. 14 (6): 595–600. arXiv:1608.00263. Bibcode:2018NatPh..14..595B. doi:10.1038/s41567-018-0124-x. S2CID 4167494.

[2] Aaronson, S. (2021). "Open Problems Related to Quantum Query Complexity". arXiv:2109.06917 [quant-ph].

[Arute19-3] Arute, F.; et al. (2019). "Quantum supremacy using a programmable superconducting processor". Nature. 574 (7779): 505–510. arXiv:1910.11333. Bibcode:2019Natur.574..505A. doi:10.1038/s41586-019-1666-5. PMID 31645734. S2CID 204836822.

[sunway-4] Liu, X.; et al. (2021). "Redefining the Quantum Supremacy Baseline With a New Generation Sunway Supercomputer". arXiv:2111.01066 [quant-ph].

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@@ Line 1: / Line 1: @@
-'''Cross-entropy benchmarking''' (also referred to as '''XEB''') is [[quantum computer|quantum]] benchmarking protocol which can be used to demonstrate [[quantum supremacy]]. <ref>{{Cite journal
+'''Cross-entropy benchmarking''' (also referred to as '''XEB''') is a [[quantum computer|quantum]] benchmarking protocol which can be used to demonstrate [[quantum supremacy]].<ref>{{Cite journal
  |last1=Boixo |first1=S.
  |display-authors=etal
@@ Line 12: / Line 12: @@
  |doi = 10.1038/s41567-018-0124-x
  |s2cid = 4167494
- }}</ref> In XEB, a random [[quantum circuit]] is executed on a quantum computer multiple times in order to collect a set of <math> k </math> samples in the form of [[bitstring|bitstrings]] <math>\{x_{1}, \dots, x_{k}\}</math>. The bitstrings are then used to calculate the cross-entropy benchmark fidelity (<math>F_{XEB}</math>) via a [[ Computer#Digital_computers| classical computer]]. <math>F_{XEB}= 2^{n} \langle P(x_{i}) \rangle_{k} - 1 =  \frac{2^{n}}{k} (\sum_{i=1}^{k}|\langle 0^{n}|C|x_{i}\rangle|^{2}) - 1</math> where <math>n</math> is the number of [[qubit|qubits]] in the circuit and <math>P(x_{i}) </math> is the probability of a bitstring <math>{x_{i}}</math> for an ideal quantum circuit <math>C</math>.  If <math>F_{XEB} = 1</math>, the samples were collected from a noiseless quantum computer. If <math>F_{XEB} = 0</math>, then the samples could have been obtained via random guessing.<ref>
+ }}</ref> In XEB, a random [[quantum circuit]] is executed on a quantum computer multiple times in order to collect a set of <math> k </math> samples in the form of [[bitstring]]s <math>\{x_{1}, \dots, x_{k}\}</math>. The bitstrings are then used to calculate the cross-entropy benchmark fidelity (<math>F_{\rm XEB}</math>) via a [[Computer#Digital computers|classical computer]], given by
+:<math>F_{\rm XEB}= 2^{n} \langle P(x_{i}) \rangle_{k} - 1 =  \frac{2^{n}}{k} \left(\sum_{i=1}^{k}|\langle 0^{n}|C|x_{i}\rangle|^{2}\right) - 1</math>,
+where <math>n</math> is the number of [[qubit]]s in the circuit and <math>P(x_{i}) </math> is the probability of a bitstring <math>{x_{i}}</math> for an ideal quantum circuit <math>C</math>.  If <math>F_{XEB} = 1</math>, the samples were collected from a noiseless quantum computer. If <math>F_{\rm XEB} = 0</math>, then the samples could have been obtained via random guessing.<ref>
  {{Cite arXiv
  |last1=Aaronson |first1=S.
@@ Line 19: / Line 21: @@
  |year = 2021
  |class=quant-ph
- }}</ref> This means that if a quantum computer did generate those samples, then the quantum computer is too noisy and thus has no chance of performing beyond-classical computations. Since it takes an exponential amount of resources to classically simulate a quantum circuit, there comes a point when the biggest supercomputer that runs the best classical algorithm for simulating quantum circuits can't compute the XEB. Crossing this point is known as achieving quantum supremacy; and after entering the quantum supremacy regime, XEB can only be estimated. <ref name="Arute19">
+ }}</ref> This means that if a quantum computer did generate those samples, then the quantum computer is too noisy and thus has no chance of performing beyond-classical computations. Since it takes an exponential amount of resources to classically simulate a quantum circuit, there comes a point when the biggest supercomputer that runs the best classical algorithm for simulating quantum circuits can't compute the XEB. Crossing this point is known as achieving quantum supremacy; and after entering the quantum supremacy regime, XEB can only be estimated.<ref name="Arute19">
 {{Cite journal
  |last1=Arute |first1=F.
@@ Line 36: / Line 38: @@
  }}</ref>
-The [[Sycamore processor]] was the first to demonstrate quantum supremacy via XEB. Instances of random circuits with <math>n = 53</math> and 20 cycles were run to obtain an XEB of <math>0.0024</math>. <ref name="Arute19"/>  Generating samples took 200 seconds on the quantum processor when it would have taken 10,000 years on [[Summit (supercomputer)|Summit]] at the time of the experiment. Improvements in classical algorithms have shortened the runtime to about a week on [[Sunway TaihuLight]] thus collapsing Sycamore's claim to quantum supremacy. <ref name="sunway">
+The [[Sycamore processor]] was the first to demonstrate quantum supremacy via XEB. Instances of random circuits with <math>n = 53</math> and 20 cycles were run to obtain an XEB of <math>0.0024</math>.<ref name="Arute19"/>  Generating samples took 200 seconds on the quantum processor when it would have taken 10,000 years on [[Summit (supercomputer)|Summit]] at the time of the experiment. Improvements in classical algorithms have shortened the runtime to about a week on [[Sunway TaihuLight]] thus collapsing Sycamore's claim to quantum supremacy.<ref name="sunway">
 {{Cite arXiv
  |last1=Liu |first1=X.
@@ Line 44: / Line 46: @@
  |eprint=2111.01066
  |class=quant-ph
- }}</ref> As of 2021, the latest demonstration of quantum supremacy by Zuchongzhi 2.1 with <math>n = 60</math>, 24 cycles and an XEB of <math>0.000366</math> holds. It takes around 4 hours to generate samples on Zuchonzhi 2.1 when it would take 10,000 years on Sunway.<ref name="sunway"/>
+ }}</ref> As of 2021, the latest demonstration of quantum supremacy by Zuchongzhi 2.1 with <math>n = 60</math>, 24 cycles and an XEB of <math>0.000366</math> holds. It takes around 4 hours to generate samples on Zuchongzhi 2.1 when it would take 10,000 years on Sunway.<ref name="sunway"/>
 == See also ==