1 Introduction
Quantum computational supremacy refers to experimental violations of the extended Church Turing Hypothesis using quantum computers. The most famous (and arguably at this point the only) example of such an experiment was carried out by Google [google19]. The Google team constructed a device that provides a “noisy simulation” of a quantum circuit with inputs and outputs. The device can be thought as a “black box” that samples from a distribution over that (loosely) approximates the distribution that corresponds to measuring applied to the allzeroes string . The quality of the device was measured using a certain benchmark known as the Linear CrossEntropy benchmark (a.k.a. Linear XEB). The computational hardness assumption underlying the experiment is that no efficient classical algorithm can achieve a similar score. In this paper we investigate this assumption, giving a new classical algorithm for “spoofing” this benchmark in certain regimes. While our algorithm falls short of spoofing the benchmark in the parameter regime corresponding to the Google experiment, we do manage to achieve nontrivial results for deeper circuits than were known before. To our knowledge, this is the first algorithm that directly targets the linear XEB benchmark, without going through a full simulation of the underlying quantum circuit. Thus our work can be viewed as evidence that obtaining nontrivial performance for this benchmark is not equivalent to simulating quantum circuits.
The linear XEB benchmark is defined as follows. Let be an qubit quantum circuit and be the pdf of the distribution obtained by measuring . For each , the instance linear XEB of is defined as
. For every probability distribution
, the linear XEB fidelity of with respect to circuit is defined asIf is a fully random circuit, then in expectation a perfect simulation achieves .^{1}^{1}1This follows since is the Porter Thomas distribution. However, is not the maximizer of : a distribution that has all its mass on the mode of the distribution will achieve for fully random circuits, and even higher values for shallower circuits as we’ll see below. Google’s “quantum computational supremacy” experiment demonstrated a noisy simulator sampling from a distribution with for two dimensional 53qubit circuits of depth 20. A trivial simulation (e.g. a distribution which is the uniform distribution or another distribution independent of ) will achieve . Motivated by the above, we say that achieves non trivial fidelity with respect to the circuit if .^{2}^{2}2As mentioned above, for an ideal simulation in random circuits the Fidelity will be a constant. For noisy quantum circuits such as Google’s, the fidelity is roughly where is the level of noise per gate and is the number of gates in the circuit.
The computational assumption underlying quantum computational supremacy with respect to some distribution over quantum circuits can be defined as follows. For every efficient randomized classical algorithm , with high probability over , if we let be the distribution of ’s output on input , then . That is, the distribution output by has trivial fidelity with respect to . Aaronson and Gunn [AG19] showed that this assumption follows from a (very strong) assumption they called “Linear CrossEntropy Quantum Threshold Assumption” or XQUATH.^{3}^{3}3While [AG19] state their result for fidelity, their proof shows that the XQUATH assumption implies that classical algorithms can not achieve empirical fidelity with samples.
In this work, we present an efficient classical algorithm that satisfies for quantum circuit sampled from a distribution with Haar random 2qubit gates with small light cones (see Definition 2.2).^{4}^{4}4If is a quantum circuit and is an output bit of , then the light cone of is the set of all input bits that are connected to via a path in the circuit. For general circuits the light cone size can be exponential in the depth, but for one or two dimensional circuits, of the type used in quantum supremacy experiment, the light cone size is polynomial in the depth. Specifically, we prove the following theorem:
Theorem 1.1 (Linear XEB for circuits with small light cones).
Let and let be a distribution over qubit quantum circuits with (i) light cone size at most , (ii) depth at most , and (iii) Haar random qubit gates. Then, there exists a classical randomized algorithm running in time such that
For constant dimensional circuits (such as the 2D quantum architecture used by Google), Theorem 1.1 yields the following corollary:
Corollary 1.2 (Constant dimensional circuits).
Let and . Let be a constant and be the distribution of qubit dimensional circuits of depth with Haar random qubit gates. There is a randomized algorithm running in time such that
Proof.
A dimensional of depth circuit has lightcone of size for . Let . By plugging in the parameters of Theorem 1.1, we see that (using and ) the expected value of the fidelity is at least
The right hand size is at least for every constant and in fact for . ∎
The bounds of Corollary 1.2 do not correspond to the Google experiment where the depth is roughly comparable to , rather than logarithmic. However, prior works in the literature were only able to achieve good linear XEB performance for circuits of constant depth (see Section 1.2). More importantly (in our view) is that our bounds show that it may be possible to achieve good linear XEB performance without achieving a full simulation.
1.1 From expectation to concentration.
In actual experiments, one measures the empirical linear XEB, obtained by sampling independently from the distribution and computing . Thus in our classical simulation we want to go beyond achieving large expected linear XEB benchmark, to show that our algorithm actually achieves nontrivial empirical linear XEB with probability at least inverse polynomial over the choice of the circuit and with a number of samples that is at most polynomial in
. These probability bounds are more challenging to prove, and at the moment our results are weaker than the optimal bounds one can hope for.
Probability over circuits.
For bounding the probability over circuits we show in Section 5.1, that in the setting of Theorem 1.1 , for logarithmic depth circuits, we can obtain fidelity with probability at least . We also obtain more general tradeoffs between the fidelity, probability, and depth, see 5.4. We conjecture that random circuits from the distributions we consider exhibit much better concentration, and fact that the fidelity sharply concentrates around its expectation.
Sample complexity, or probability over the algorithms’ randomness.
Bounding the sample complexity of our algorithm is a more difficult task then the expectation analysis because it requires higher moment information on . We obtain only partial bounds in this setting, which we believe to be far from optimal. In Section 6 we show that an upper bound for the collision probability of is sufficient to give an upper bound for the sample complexity of our algorithm. Specifically, letting , we show that if then the number of samples needed for the empirical linear XEB to achieve a value of at least is . In particular, for logarithmic depth circuits we can get inversepolynomial empirical fidelity using samples. For random quantum circuits, where is the PorterThomas distribution (with
drawn independently as the square of a mean zero variance
normal variable), it is known that , i.e., . For shallow circuits, of the type we study, we show in 6.6 that for random one dimensional circuits of depth at least for some constant , which shows that we can achieve for such circuits inverse polynomial empirical fidelity using a polynomial number of samples. While this is significantly more technically challenging to prove, we conjecture that the same collision probability bound holds for two dimensional circuits of depth . This conjecture, if true, will imply that for such circuits we can achieve empirical fidelity using a polynomial number of samples, and constant fidelity using a subexponential (e.g. ) number of samples.1.2 Prior works
Prior classical algorithms mostly focused on the task of obtaining a full simulation (sampling from or from a distribution close to it in statistical distance). We are not aware of any prior work that directly targeted the linear XEB measure and gave explicit bounds for the performance in this measure that are not implied by approximating the full distribution.
Napp et al [napp2019efficient] gave an algorithm to simulate random twodimensional circuits of some small constant depth. They gave strong theoretical evidence that up to a certain constant depth, such circuits can be approximated by 1D circuits of small entanglement (i.e., “area law” as opposed to “volume law”), which can be effectively simulated using Matrix Product States [vidal2004efficient]. However, [napp2019efficient] also gave evidence that the system undergoes a phase transition when the depth is more than some constant size (around ), at which case the entanglement grows according to a “volume law” and hence their methods cannot be used to simulate circuits of superconstant depth.
Another direction of approximating large quantum circuits has considered the effect of noise. Some restricted classes of noisy quantum circuits were shown to be simulated by polynomial time classical algorithms in [bremner2017achieving, yung2017can] (in contrast to their noiseless variant [bremner2016average]). This was also extended to more general random circuits by [gao2018efficient]. Very recent work has given numerical results suggesting that states generated by noisy quantum circuits could be approximated by Matrix Product States or Operators under the state fidelity measure [zhou2020limits, noh2020efficient].^{5}^{5}5[zhou2020limits] briefly discusses the linear XEB measure as well, see Figure 7 there. Low degree Fourier expansions yield other candidates for approximating such quantum states [gao2018efficient, bremner2017achieving].
2 Preliminaries
In this section we introduce some of the notions we use for quantum circuits, and in particular distributions of random quantum circuits of fixed architecture, as well as tensor networks for analyzing quantum circuits. We include this here since some of this notation, and in particular tensor networks, might be unfamiliar to theoretical computer science audience. However, the reader can choose to skip this section and refer back to it as needed. We also record some useful facts of tensor networks and quantum circuits in
Appendix A.The following definition captures the notion of an “architecture” of a quantum circuit (see Figure 1 for an example):
Definition 2.1 (Circuit skeleton and light cone).
Let and , an qubit depth circuit skeleton is a directed acyclic graph with layers with the following structure. For convenience, we start the index of layers from .

The and the layer has nodes corresponding to the input and output qubits. Each node in the first layer has exactly one outgoing edge to the next layer while each node the last layer has exactly one ingoing edge from the previous layer.

Each of the other layers has exactly nodes and each node has exactly two ingoing to the next layer and two outgoing edges from the previous layer. Specifically, the first edge gate is indexed by while the second edge is indexed by for each .

For each , the input node connects to the edge of the second layer while the edge of the layer connects to the output node.
Note that with the above definition, a circuit skeleton can be specified by many permutations . Namely, for each and , the edge of the layer connects to the edge of the layer.
For every circuit skeleton , the light cone size of is the maximum over all output qubits of the size of the set .
Next, we define the light cone for an output qubit and the light cone size for a circuit skeleton.
Definition 2.2 (Light cone).
Let be a circuit skeleton and be an output qubit. The light cone of is the set of all input vertices in that has a path from left to right that ends at . The light cone size of is then defined as the largest light cone size of an output qubit in .
Note that the light cone size of the 1D circuit in Figure 1 is , which is less than the number of qubits. Also, it turns out that computing the marginal of an output qubit only requires the information from the light cone.
Lemma 2.3 (Marginal probability and light cone).
Let be a circuit skeleton with light cone size and be a circuit using skeleton . For each output qubit of , the marginal probability can be computed in time .
Proof of 2.3.
We use the circuit skeleton in Figure 1 as an illustrating example. For an output qubit in , to compute its marginal probability it suffices to compute the input state to the gate it connects to. For example, for output qubit , it suffices to compute the input state to gate (a).
Similarly, to compute the input state of a gate, it suffices to compute the input states of the gate it connects to from the previous layer. Namely, to compute the input state of gate (a), it suffices to compute that of gate (b) and (c). If we continue this process inductively, the only input state needed to compute the marginal probability of an output bit is then the one lies in its light cone. In this example, to compute the marginal probability of , it suffices to consider only .
Finally, to compute the input states of all the intermediate gates, it suffices to perform matrix vector multiplication because each intermediate state is of size at most . While all the above operations can be done in times, computing the marginal probability of an output qubits in only requires time. ∎
Now, we are able to formally define random quantum circuits.
Definition 2.4 (Random quantum circuits).
Let , . A distribution of qubit depth random circuits consists of an qubit depth circuit skeleton and ensembles over unitary matrices for each and .
A random quantum circuit sampled from by sampling a unitary matrix from and assigning to the node of the layer for each and .
Specifically, if each is Haar random, then we say is Haar random qubit circuits over .
2.1 Tensor networks
Tensor network is an intuitive graphical language that can be rigorously used in reasoning about multilinear maps. Especially, it finds many applications in quantum computing since the basic operations such as partial measurement are all multilinear maps. In this paper, we restrict our attention to qubits (as opposed to the general case of qudits) and only to gates that act on two qubits.
In a tensor network, we represent a unitary matrix (e.g., a gate) as a box with lines on the sides (see Figure 2). Each line represents a coordinate of the gate and in this paper each coordinate has dimension and is indexed by . Specifically, a line on the left represents a column vector (i.e., ) while a line on the right represents a row vector (i.e., ).^{6}^{6}6This is when the tensor network is written left to right  sometimes it is written top to bottom, in which case a line on the top represents a column vector and a line on the bottom represents a row vector. For example, Figure 2 represents .
Similarly, a state (e.g., a qubit)is represented by a triangle with line only on one side and it is a (resp. ) if the freeend of the line is left (resp. right). For example, Figure 2 represents .
Semantically, a pure line refers to an indicator function^{7}^{7}7Also known as contraction. for its two ends. For example, the line in Figure 2 reads as where if ; otherwise it is .
3 Our Algorithm
We now describe our classical algorithm that spoofs the linear crossentropy benchmark in shallow quantum circuits. The key idea is that rather than directly simulating the whole quantum circuit, our algorithm only computes the marginal distributions of few output qubits and then samples substrings for those qubit accordingly. We sample the remaining subits uniformly at random. Intuitively, due to the correlation on those output qubits, one can expect that the linear crossentropy of our algorithm could be better than uniform distribution, but the analysis is somewhat delicate. Because consider shallow quantum circuits (of at most logarithmic light cone size), the marginal of few output qubits can be efficiently computed.
Running time of the algorithm.
The total running time of Algorithm 1 is at most . Finding outputs with disjoint light cones takes time by a greedy algorithm. The second step takes time because it suffices to keep track of the density matrix recording the marginal probability of every qubit in the light cone of for each (see 2.3). The final step of sampling uniform bits for the remaining outputs can be done in polynomial time.
3.1 Analysis
The following theorem implies Theorem 1.1 by setting :
Theorem 3.1 (Linear XEB for circuits with small light cones.).
Let and let be a distribution over qubit quantum circuits with (i) light cone size at most , (ii) depth at most , and (iii) Haar random qubit gates. Then, letting be the distribution output by Algorithm 1 on input ,
where is the parameter chosen in step 1 of the algorithm.
The proof of Theorem 3.1 consists of three steps:

We reduce analyzing analyzing the expectation when the algorithms samples the marginals of output qubits into analyzing it for a single output qubit.

We apply the integration formula for Haar measure and rewrite the expected linear XEB of a single qubit into a tensor network.

We then perform a change of basis on the tensor network and turn the single qubit analysis into a Markov chain problem where the expected linear XEB of a single qubit can be easily lower bounded.
Since the heart of the proof is the single output qubit analysis, we will describe it first.
4 Single qubit analysis
In this section, we prove the case of our algorithm. That is, we prove that for a single output qubit, the expected contribution to linear XEB is of the order of .
Theorem 4.1 (Linear XEB of a single output qubit).
Let and be distribution over qubit quantum circuits with depth at most and with Haar random qubit gates. For , let denote the unitary matrix computed by . For each , we have
where .
We prove Theorem 4.1 by reducing to a Markov chain problem using tensor networks. Without loss of generality we can assume . Also, by A.1,
So our goal is to show that
(4.2) 
Let us start with rewriting the trace term of Equation 4.2 into an equivalent tensor network expression as follows.
Next, for a single gate in a quantum circuit, its expected behavior over the choice of qubit Haar random gates can be characterized in the following lemma.
Lemma 4.3.
Let be Haar random qubit gate, then the following holds.
where
The proof of 4.3 is based on the integration formula [BB96] for Haar measure. We postpone the proof of 4.3 to Section 4.1. Intuitively, the lemma says that by a change a basis, the expected behavior of a single Haar random qubit gate can be exactly understood by an explicit transition matrix . By the linearity of taking expectation, we can apply 4.3 on every gates in the circuit and thus the whole tensor network is simplified to a Markov chain. Concretely, we have the following lemma.
Lemma 4.4 (Rewrite Equation 4.2 as a Markov chain).
Let and be a Haar random distribution over an qubit depth circuit skeleton with permutations . For , let denote the unitary matrix computed by .
(4.5) 
where
The proof of 4.4 is based on a careful composition of applying 4.3 on each of the gates. We postpone the proof of 4.4 to Section 4.2. Now, we are ready to prove Theorem 4.1 and complete the analysis for the expected linear XEB of single output qubit.
Proof of Theorem 4.1.
4.4 rewrites the desiring quantity into the form of a Markov chain so now it suffices to show that the right hand side of Equation 4.5 is at least .
Notice that for every possible assignment to , . That is, it suffices to find an assignment such that . Specifically, let us consider the following assignment. For all and , let
To analyze this assignment, let us start with the last layer. There we have and for each and thus
Next, for each and , observe that due to the choice of the assignment. As a result, all the will be either or . Specifically, for each , since there is exactly one appears among while the rest are s, there is also exactly one term contributes in while the other terms are . Namely, we have
for each .
Finally, since there is exactly one appears in while the rest are s, we have
To sum up, we conclude that as desired. Specifically, this implies Equation 4.2, i.e., . Combine with A.1, this completes the proof of Theorem 4.1. ∎
4.1 Proof of Lemma 4.3
We start with applying the integration formula for Haar random matrix and considering its tensor netowrok representation.
Lemma 4.6 ([Bb96, Equation 2.4]).
Let be a Haar random qubit gate, then we have the following. For each ,
The above equation can be represented as the following tensor network.
Next, the idea is to apply the Pauli identity (i.e., Equation A.3) on each pair of ,, , , , , , and . Intuitively, this is doing a change of basis from the standard basis to Pauli basis.
Let us first apply the Pauli identity on and note that by A.4, we have
tikz/paulionecircle 
(4.7)  
and  
tikz/paulitwocircles 
(4.8) 
That is, the tensor network is nonzero only if . Thus, we only need one variable to handle and . Similarly, we can use to handle other pairs respectively. The equation becomes the following.
To finish the proof of 4.3, we have to explicitly calculate the value of the tensor network for each choice of . Again, by Equation 4.7 and Equation 4.8, we have the following observations.

If , then the value is .

If and at least one of is not , or at least one of is not and , then the value is .

For all the other cases, the value is .
Finally, we take out the and evenly distribute it to the Pauli gates outside. Namely, each of them gets an extra factor as shown in the equation. This completes the proof of 4.3.
4.2 Proof of Lemma 4.4
Let us do a change of basis from the standard basis to the Pauli basis. Concretely, we apply 4.3 on every gate. Note that by the independence of each gate and the linearity of expectation, the layer of the circuit becomes the following for each .
Next, the output wire at the layer, i.e., the wires indexed by , connects to the input wire at the later, i.e., the wires indexed by . By the orthogonality of Pauli gates (i.e., A.4), we have for all . To sum up, the to layer is equivalent to following.
Finally, let us plug in the input and output layer. Recall that the input layer contains 4 copies of and the output layer contains 2 copies of . Concretely, the contribution from the input layer would be
while the contribution from the output layer would be
This completes the proof of 4.4.
5 Wrapping up: from single output bit to many bits
In this section we complete the proof of Theorem 1.1 .
We will use the following notation. Let be a pdf over . For any and , let denote the marginal probability of the output qubit at location being . Formally, . For a fixed input , let be the output qubits selected by Algorithm 1. Note that Algorithm 1 will choose the same for each sampled from . By the design of Algorithm 1, for every . Thus, the linear XEB of is the following.
(5.1)  
Note that because their light cones are disjoint, by A.5, we have . Thus, the equation becomes  
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