Classically Simulating Quantum Circuits with Local Depolarizing Noise

We study the effect of noise on the classical simulatability of quantum circuits defined by computationally tractable (CT) states and efficiently computable sparse (ECS) operations. Examples of such circuits, which we call CT-ECS circuits, are IQP, Clifford Magic, and conjugated Clifford circuits. This means that there exist various CT-ECS circuits such that their output probability distributions are anti-concentrated and not classically simulatable in the noise-free setting (under plausible assumptions). First, we consider a noise model where a depolarizing channel with an arbitrarily small constant rate is applied to each qubit at the end of computation. We show that, under this noise model, if an approximate value of the noise rate is known, any CT-ECS circuit with an anti-concentrated output probability distribution is classically simulatable. This indicates that the presence of small noise drastically affects the classical simulatability of CT-ECS circuits. Then, we consider an extension of the noise model where the noise rate can vary with each qubit, and provide a similar sufficient condition for classically simulating CT-ECS circuits with anti-concentrated output probability distributions.

Authors

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1 Introduction

1.1 Background and Main Results

A key step toward realizing a large-scale universal quantum computer is to demonstrate quantum computational supremacy [11], i.e., to perform computational tasks that are classically hard. As such a task, many researchers have focused on simulating quantum circuits, or more concretely, sampling the output probability distributions of quantum circuits. They have shown that, under plausible complexity-theoretic assumptions, this task is classically hard for various quantum circuits that seem easier to implement than universal ones. However, these classical hardness results have been obtained in severely restricted settings, such as a noise-free setting with additive approximation [5, 18, 3, 21] and a noise setting with multiplicative approximation [9]: the former requires us to sample the output probability distribution of a quantum circuit with additive error and the latter to sample the output probability distribution of a quantum circuit under a noise model with multiplicative error. Thus, there is great interest in considering the above task in a more reasonable setting.

We study the classical simulatability of quantum circuits in a noise setting with additive approximation, which requires us to sample the output probability distribution of a quantum circuit under a noise model with additive error. This setting is more reasonable than the noise-free setting with additive approximation since the presence of noise is unavoidable in realistic situations. Moreover, our setting is more reasonable than a noise setting with multiplicative approximation in the sense that we adopt a more realistic notion of approximation [1, 5], although noise in this paper is more restrictive than that in [9]. We consider a noise model where a depolarizing channel with an arbitrarily small constant rate , which is denoted as , is applied to each qubit at the end of computation. This channel leaves a qubit unaffected with probability and replaces its state with the completely mixed one with probability . We call this model noise model A. We also consider its extension where the noise rate can vary with each qubit. More concretely, when a quantum circuit has qubits, is applied to the -th qubit at the end of computation for any . We call this model noise model B. These noise models are simple, but analyzing them is a meaningful step toward studying more general models [10], such as one where noise exists before and after each gate in a quantum circuit. This is because, for example, this general noise model is equivalent to noise model A when we focus on instantaneous quantum polynomial-time (IQP) circuits, which are described below, with a particular type of intermediate noise [6].

A representative example of a quantum circuit that is not classically simulatable (in the noise-free setting) is an IQP circuit, which consists of -diagonal gates sandwiched by two Hadamard layers. In fact, there exists an IQP circuit such that its output probability distribution is anti-concentrated and not classically samplable in polynomial time with certain constant accuracy in norm (under plausible assumptions) [5]. On the other hand, Bremner et al. [6] studied the classical simulatability of IQP circuits under noise model A.111Bremner et al. also dealt with quantum circuits for Simon’s algorithm. Our results can be directly extended to such circuits with access to an oracle, although we omit the details for simplicity. They showed that, if the exact value of the noise rate is known, any IQP circuit with an anti-concentrated output probability distribution is classically simulatable in the sense that the resulting probability distribution is classically samplable in polynomial time with arbitrary constant accuracy in norm. This indicates that, under noise model A, if the exact value of the noise rate is known, the presence of small noise drastically affects the classical simulatability of IQP circuits.

In this paper, first, under a weaker assumption on the knowledge of the noise rate, we extend Bremner et al.’s result to quantum circuits that are defined by two concepts: computationally tractable (CT) states and efficiently computable sparse (ECS) operations [20]. Examples of such circuits, which we call CT-ECS circuits, are IQP circuits, Clifford Magic circuits [21], and conjugated Clifford circuits [3]. This means that there exist various CT-ECS circuits such that their output probability distributions are anti-concentrated and not classically simulatable in the sense described above for IQP circuits (under plausible assumptions). Constant-depth quantum circuits [19, 4, 2] are also CT-ECS circuits and not classically simulatable, although we do not know whether their output probability distributions are anti-concentrated. We postpone the explanation of CT states and ECS operations until Section 2, but, as depicted in Fig. 1(a), a CT-ECS circuit on qubits is a polynomial-size quantum circuit such that is CT and is ECS for any , where is a Pauli- operation on the -th qubit. After performing , we perform -basis measurements on all qubits. The CT-ECS circuit under noise model B is depicted in Fig. 1(b).

Our first result assumes noise model A, which corresponds to the case where for any in Fig. 1(b). We show that, if an approximate value of the noise rate is known, any CT-ECS circuit with an anti-concentrated output probability distribution is classically simulatable:

Theorem 1 (informal).

Let be an arbitrary CT-ECS circuit on qubits such that its output probability distribution is anti-concentrated, i.e., for some known constant . We assume that a depolarizing channel with (possibly unknown) constant rate is applied to each qubit after performing , which yields the probability distribution . Moreover, we assume that it is possible to choose a constant such that

 1≤ελ≤1+c,

where is a certain constant depending on . Then, is classically samplable in polynomial time with constant accuracy in norm.

Throughout the paper, the base of the logarithm is 2. If is known, we can choose in Theorem 1. This case with IQP circuits precisely corresponds to Bremner et al.’s result [6]. As described above, there exist various CT-ECS circuits such that their output probability distributions are anti-concentrated and not classically simulatable in the noise-free setting (under plausible assumptions). Thus, Theorem 1 indicates that, under noise model A, if an approximate value of the noise rate is known, the presence of small noise drastically affects the classical simulatability of CT-ECS circuits.

Theorem 1 assumes noise model A where noise exists only at the end of computation, but, in some cases, it can be applied to an input-noise model. For example, Theorem 1 holds for IQP circuits when noise model A is replaced with a noise model where is applied to each qubit only at the start of computation, although Bu et al.’s main result implies a similar property of IQP circuits [7]. Moreover, Theorem 2, which is described below and assumes noise model B, also holds for IQP circuits when noise model B is replaced with an input-noise model where the noise rate can vary with each qubit. Our main result is similar in spirit to Bu et al.’s, which provides classical algorithms for simulating Clifford circuits with nonstabilizer product input states (corresponding to input-noise models). However, we note that, in general, it is difficult to relate the output probability distributions of CT-ECS circuits under output-noise models to those of Clifford circuits under input-noise models.

Our main focus is on a noise setting, but, from a purely theoretical point of view, it is valuable to analyze the classical simulatability of quantum circuits in the noise-free setting. The proof method of Theorem 1 is based on computing the Fourier coefficients of an output probability distribution, and is useful in the noise-free setting. In fact, it implies that, when only qubits are measured, any quantum circuit in a class of CT-ECS circuits on qubits is classically simulatable. More precisely, its output probability distribution is classically samplable in polynomial time with polynomial accuracy in norm. This class of CT-ECS circuits is defined by a restricted version of ECS operations, and includes IQP, Clifford Magic, conjugated Clifford, and constant-depth quantum circuits. It is known that the above property or a similar one holds for these quantum circuits (although the notions of approximation vary), but the proofs provided have depended on each circuit class [19, 4, 12, 3]. Our analysis unifies the previous ones and clarifies a class of quantum circuits for which the above property holds.

Our second result assumes noise model B, which is depicted in Fig. 1(b). For classically simulating CT-ECS circuits with anti-concentrated output probability distributions, we provide a sufficient condition, which is similar to Theorem 1:

Theorem 2 (informal).

Let be an arbitrary CT-ECS circuit on qubits such that its output probability distribution satisfies for some known constant . We assume that a depolarizing channel with (possibly unknown) constant rate is applied to the -th qubit after performing for any , which yields the probability distribution . Moreover, we assume that it is possible to choose a constant such that

 1≤εminλmin≤1+c,

where and is a certain constant depending on , and we assume that it is possible to choose a constant such that

 0≤εj−λj≤cλmin

for any with . Then, is classically samplable in polynomial time with constant accuracy in norm.

To the best of our knowledge, this is the first analysis of the classical simulatability of quantum circuits under noise model B. Theorem 2 indicates that, under this noise model, if approximate values of the minimum noise rate and the other noise rates are known, the presence of small noise drastically affects the classical simulatability of CT-ECS circuits.

1.2 Overview of Techniques

To prove Theorem 1, we generalize Bremner et al.’s proof for IQP circuits [6]. There are two key points. The first one is to provide a general method for approximating the Fourier coefficients of the output probability distribution . It is known that the probability distribution , which we want to approximate, can be simply represented by the noise rate and the Fourier coefficients of for all  [15]. We show that, for any CT-ECS circuit on qubits, there exists a polynomial-time classical algorithm for approximating each of the low-degree Fourier coefficients of , i.e., for any with . Bremner et al. [6] showed that such an algorithm exists for IQP circuits through a direct calculation of the Fourier coefficients for them. In contrast, we first provide a general relation between a quantum circuit and the Fourier coefficients of its output probability distribution. We then approximate each of the low-degree Fourier coefficients by combining this general relation with Nest’s classical algorithm for approximating the inner product value of a particular form defined by a CT state and an ECS operation [20]. This general relation seems to be a new tool to investigate the output probability distribution of a quantum circuit and thus may be of independent interest.

The second key point is to approximate using an approximate value of . We define a function that seems to be close to on the basis of its representation with and the Fourier coefficients for all . Unfortunately, in contrast to Bremner et al.’s setting, we do not know . Thus, using an approximate value of , we first choose an appropriate (polynomial) number of the low-degree Fourier coefficients used in , and define based on the above representation of . More precisely, this number depends on , the constant associated with the anti-concentration assumption, and the desired approximation accuracy. We then evaluate the approximation accuracy of . Here, we need to care about the error caused by the difference between and , and we upper-bound this error using the anti-concentration assumption.

To prove Theorem 2, we represent the effect of noise under noise model B as the effect of noise under noise model A with rate and the remaining effects. We do this by transforming the representation of the probability distribution , which we want to approximate, with several basic properties of noise operators on real-valued functions over  [15]. The obtained representation means that, to sample , it suffices to sample the probability distribution (resulting from ) under noise model A with rate and then to classically simulate noise corresponding to the remaining effects. By Theorem 1, is classically samplable in polynomial time with arbitrary constant accuracy in norm. Moreover, we can simulate noise corresponding to the remaining effects using the approximate values of and ’s, which are not equal to .

2 Preliminaries

2.1 Quantum Circuits and Their Output Probability Distributions

Pauli matrices , , , and are

 X=(0110), Y=(0−ii0), Z=(100−1), I=(1001).

The Hadamard operation is defined as . For any real number , the rotation operations and are defined as

 Rx(θ)=cosθ2I−isinθ2X, Rz(θ)=cosθ2I−isinθ2Z.

In particular, and are denoted as and , respectively. It is easy to verify that the inverse of is and, similarly, the inverse of is . The controlled- operation and the controlled-controlled- operation are defined as

 CZ=|0⟩⟨0|⊗I+|1⟩⟨1|⊗Z, CCZ=|00⟩⟨00|⊗I+|01⟩⟨01|⊗I+|10⟩⟨10|⊗I+|11⟩⟨11|⊗Z,

where the states and are -eigenstates of , respectively. The operations , , and are called Clifford operations.

A quantum circuit consists of elementary gates, each of which is in the gate set . Here, . Each has its inverse in and so does . Moreover, each of the one-qubit operations , , , , and can be decomposed into a constant number of ’s and ’s (up to an unimportant global phase). Similarly, can be decomposed into a constant number of ’s, ’s, and ’s. Since the gate set is approximately universal for quantum computation [13], so is . The complexity measures of a quantum circuit are its size and depth. The size of a quantum circuit is the number of elementary gates in the circuit. To define the depth, we regard the circuit as a set of layers consisting of elementary gates, where gates in the same layer act on pairwise disjoint sets of qubits and any gate in layer is applied before any gate in layer . The depth is defined as the smallest possible value of  [8].

We deal with a (polynomial-time) uniform family of polynomial-size quantum circuits . Each has input qubits initialized to . After performing , we perform -basis measurements on all qubits. The output probability distribution of over is defined as . A symbol denoting a quantum circuit also denotes its matrix representation in the basis. The (polynomial-time) uniformity means that the function

is computable by a polynomial-time classical Turing machine, where

is the classical description of  [14].

2.2 Fourier Expansions and Effects of Noise

Let be an arbitrary (real-valued) function. Then, can be uniquely represented as an -linear combination of basis functions

 f(x)=∑s∈{0,1}nˆf(s)(−1)s⋅x,

which is called the Fourier expansion of  [15]. Here, is called the Fourier coefficient of and the symbol “” represents the inner product of two -bit strings, i.e., for any , . It holds that

 ˆf(s)=12n∑x∈{0,1}nf(x)(−1)s⋅x

for any . The norm of is defined as

 ||f||k=⎛⎝∑x∈{0,1}n|f(x)|k⎞⎠1/k,

where . It is known that  [15].

Let be an arbitrary quantum circuit on qubits and be its output probability distribution over . We consider under noise model A where a depolarizing channel with rate is applied to each qubit after performing . Here, for any density operator of a qubit. We perform -basis measurements on all qubits and let be the resulting probability distribution over . As shown in [6], we can sample by sampling an -bit string according to and then flipping each bit of the string with probability . This implies the following Fourier expansion of  [15]:

 ˜pA(x)=∑s∈{0,1}n(1−ε)|s|ˆp(s)(−1)s⋅x,

where for any . We also consider under noise model B where with rate is applied to the -th qubit after performing for any . We perform -basis measurements on all qubits and let be the resulting probability distribution over . As with , we can sample by sampling an -bit string according to and then flipping its -th bit with probability for any . The Fourier expansion of is as follows [15]:

 ˜pB(x)=∑s∈{0,1}n[n∏j=1(1−εj)sj]ˆp(s)(−1)s⋅x.

2.3 CT States and ECS Operations

We introduce CT states and (a restricted version of) ECS operations [20]. Let be an arbitrary (pure) quantum state on qubits and be the probability distribution over defined as . Then, is CT if is classically samplable in polynomial time and, for any , is classically computable in polynomial time.222For simplicity, we require perfect accuracy in sampling probability distributions and computing values, although irrational numbers may be involved. Precisely speaking, it suffices to require exponential accuracy. This is also applied to similar situations in this paper, such as the definition of ECS operations. An example of a CT state is a product state.

Let be an arbitrary quantum operation on qubits that is both unitary and Hermitian. The operation is sparse if there exists a polynomial in such that, for any , is a linear combination of at most computational basis states. When is a sparse operation (associated with ), for any , we define two functions, and , as follows: for any , if the -th non-zero entry exists in the column indexed by when traversing this column from top to bottom, is this entry and is the row index associated with . If the -th non-zero entry does not exist in this column, and . The sparse operation is ECS if, for any and , and are classically computable in polynomial time. In particular, an ECS operation with is called ECS. Moreover, an ECS operation with is called efficiently computable basis-preserving.

An efficiently computable basis-preserving operation preserves the class of CT states [20]:

Theorem 3 ([20]).

Let be an arbitrary CT state on qubits and be an arbitrary efficiently computable basis-preserving operation on qubits. Then, is CT.

The following theorem is a rephrased version of the one in [20]:

Theorem 4 ([20]).

Let be an arbitrary quantum operation on qubits such that is CT, and be an arbitrary observable with , where

is the absolute value of the largest eigenvalue of

. Let be an arbitrary quantum operation on qubits such that is ECS, and be an arbitrary polynomial in . Then, there exists a polynomial-time randomized algorithm which outputs a real number such that

 Pr[∣∣⟨0n|U†V†OVU|0n⟩−r∣∣≤1f(n)]≥1−1exp(n).

3 Target Quantum Circuits and Associated Fourier Coefficients

3.1 CT-ECS Circuits

We focus on a new class of quantum circuits defined by CT states and ECS operations:

Definition 1.

A quantum circuit on qubits initialized to is CT-ECS if consists of two blocks such that and are polynomial-size quantum circuits, is CT, and is ECS for any , where is a Pauli- operation on the -th qubit.

To provide examples of CT-ECS circuits, we define IQP, Clifford Magic, conjugated Clifford, and constant-depth quantum circuits on qubits as follows:

• An IQP circuit is of the form , where is a polynomial-size quantum circuit consisting of , , and gates [5].

• A Clifford Magic circuit is of the form , where is a polynomial-size Clifford circuit, which consists of , , and gates [21].

• A conjugated Clifford circuit is of the form for arbitrary real numbers , where is a polynomial-size Clifford circuit [3].

• A constant-depth quantum circuit is a polynomial-size quantum circuit whose depth is constant [19, 4, 2].

In the common definition of an IQP circuit, consists of more general -diagonal gates. However, for simplicity, we adopt the above definition. The resulting class includes a quantum circuit of the form such that its output probability distribution is anti-concentrated and not classically simulatable (under plausible assumptions).

We show that the above circuits are CT-ECS:

Lemma 1.

Let be one of the following quantum circuits on qubits: an IQP, a Clifford Magic, a conjugated Clifford, or a constant-depth quantum circuit. Then, is CT-ECS.

Proof.

When is an IQP circuit, , where is a polynomial-size quantum circuit consisting of , , and gates. We consider and . Since is a product state and is an efficiently computable basis-preserving operation, by Theorem 3, is CT. Moreover, , which is obviously ECS (in fact, efficiently computable basis-preserving), where is a Pauli- operation on the -th qubit. Thus, is CT-ECS.

When is a Clifford Magic circuit, , where is a polynomial-size Clifford circuit. We consider and . Since is a product state, it is CT. Moreover, since is a Clifford circuit, is a Pauli operation on qubits, which is obviously ECS (in fact, efficiently computable basis-preserving). Thus, is CT-ECS.

When is a conjugated Clifford circuit, for arbitrary real numbers , where is a polynomial-size Clifford circuit. We consider and . Since is a product state, it is CT. Moreover,

 V†ZjV=E†Rz(ϕ)jRx(θ)jZjRx(−θ)jRz(−ϕ)jE.

The coefficients satisfying the following relation are classically computable in constant time:

 α1I+α2Z+α3X+α4Y=Rz(ϕ)jRx(θ)jZjRx(−θ)jRz(−ϕ)j.

This implies that . Since is a Clifford circuit, the operations , , and are Pauli operations on qubits, which implies that is ECS. Thus, is CT-ECS.

When is a constant-depth quantum circuit, we consider and . Of course, is CT. Moreover, since each elementary gate in this paper acts only on a constant number of qubits, also acts only on a constant number of qubits, which implies that is ECS. Thus, is CT-ECS. ∎

Lemma 1 implies that there exist various CT-ECS circuits such that their output probability distributions are anti-concentrated and not classically simulatable (under plausible assumptions), although we do not know whether the output probability distributions of constant-depth quantum circuits are anti-concentrated.

The above proof implies that, for any quantum circuit in Lemma 1, the associated ECS operation satisfies the condition where, for any , can be represented as a linear combination of at most computational basis states. In other words, is ECS. This defines a class of CT-ECS circuits, whose elements we call CT-ECS circuits. In Section 4.3, we consider the classical simulatability of CT-ECS circuits on qubits in the noise-free setting when only qubits are measured.

3.2 Approximating the Associated Fourier Coefficients

We provide a general relation between a quantum circuit and the Fourier coefficients of its output probability distribution:

Lemma 2.

Let be an arbitrary quantum circuit on qubits initialized to and be its output probability distribution over . Then,

 ˆp(s)=12n⟨0n|C†ZsC|0n⟩

for any , where

, i.e., the tensor product of a Pauli-

operation on the -th qubit with for any .

Proof.

We transform the representation of the Fourier coefficient described in Section 2.2 as follows:

 ˆp(s) =12n∑x∈{0,1}np(x)(−1)s⋅x=12n∑x∈{0,1}n⟨0n|C†|x⟩⟨x|C|0n⟩(−1)s⋅x =12n∑x∈{0,1}n⟨0n|C†Xx|0n⟩⟨0n|XxC|0n⟩(−1)s⋅x, (1)

where denotes for any . Since it holds that

 |0n⟩⟨0n|=12n∑t∈{0,1}nZt,

the above representation (3.2) of implies that

 ˆp(s) =122n∑x∈{0,1}n∑t∈{0,1}n⟨0n|C†XxZtXxC|0n⟩(−1)s⋅x =122n∑t∈{0,1}n⟨0n|C†ZtC|0n⟩∑x∈{0,1}n(−1)(s+t)⋅x=12n⟨0n|C†ZsC|0n⟩,

where is the bit-wise addition of and modulo 2. This is the desired representation. ∎

Using Theorem 4 and Lemma 2, we show that the low-degree Fourier coefficients of the output probability distribution of a CT-ECS circuit can be approximated classically in polynomial time:

Lemma 3.

Let be an arbitrary CT-ECS circuit on qubits and be its output probability distribution over . Let be an arbitrary polynomial in and be an arbitrary element of with . Then, there exists a polynomial-time randomized algorithm which outputs a real number such that

 Pr[|ˆp(s)−ˆp′(s)|≤12nf(n)]≥1−1exp(n).
Proof.

Since is CT-ECS, it can be represented as such that is CT and is ECS for any . Let be an arbitrary element of with , and we assume that . In this case,

 V†ZsV=|s|∏k=1(V†ZjkV).

Since is ECS, is the product of a constant number of ECS operations. A simple calculation shows that such a product is also ECS. Thus, is ECS.

Since is CT and , by Theorem 4, there exists a polynomial-time randomized algorithm which outputs a real number such that

 Pr[∣∣⟨0n|U†V†ZsVU|0n⟩−r(s)∣∣≤1f(n)]≥1−1exp(n).

By Lemma 2 with ,

 ˆp(s)=12n⟨0n|U†V†ZsVU|0n⟩

and thus the desired relation holds by defining . ∎

For any probability distribution over , it holds that

 ˆp(0n)=12n∑x∈{0,1}np(x)=12n.

Thus, when we consider a classical algorithm for approximating , we only consider an algorithm that outputs when . This slightly simplifies the analysis of the classical simulatability of CT-ECS circuits in the following sections.

4 CT-ECS Circuits under Noise Model A

In this section, we prove Theorem 1. Its precise statement is as follows:

Theorem 1.

Let be an arbitrary CT-ECS circuit on qubits such that its output probability distribution over is anti-concentrated, i.e., for some known constant . Let be an arbitrary constant. We assume that

• a depolarizing channel with (possibly unknown) constant rate is applied to each qubit after performing , which yields the probability distribution over , and

• it is possible to choose a constant such that

 1≤ελ≤1+110√αδlog10√αδ.

Then, there exists a polynomial-time randomized algorithm which outputs (a classical description of) a probability distribution over such that

 Pr[||˜pA−˜qA||1≤δ]≥1−1exp(n)

and is classically samplable in polynomial time.

First, we define a function over that is close to , which we want to approximate. This is done by using the approximate value of the noise rate and the approximate values of the low-degree Fourier coefficients of obtained by Lemma 3. Then, we sample a probability distribution close to the function.

4.1 Function Close to the Target Probability Distribution

We show that the probability distribution can be approximated by a function whose Fourier coefficients can be obtained classically in polynomial time:

Lemma 4.

Let be an arbitrary CT-ECS circuit on qubits such that its output probability distribution satisfies for some known constant . Let be an arbitrary constant. We assume that

• a depolarizing channel with constant rate is applied to each qubit after performing , which yields the probability distribution , and

• it is possible to choose a constant such that

 1≤ελ≤1+110√αδlog10√αδ.

Then, there exists a polynomial-time randomized algorithm which outputs the Fourier coefficients of a function over such that

 Pr[||˜pA−q||1≤δ3]≥1−1exp(n).
Proof.

As described in Section 2.2, the probability distribution is represented as

 ˜pA(x)=∑s∈{0,1}n(1−ε)|s|ˆp(s)(−1)s⋅x.

Using the known constants , , and , we fix an integer constant

 c=⌈1λlog10√αδ⌉.

Since and , . The definition of implies that

 1λlog10√αδ≤c≤1λlog10√αδ+1≤2λlog10√αδ.

Thus, . Moreover, since

 λ≤ε≤λ+λ10√αδlog10√αδ≤λ+λ10√αδ⋅cλ2≤λ+δ5c√α.

By Lemma 3 with and an arbitrary with , there exists a polynomial-time randomized algorithm which outputs such that

 Pr[|ˆp(s)−ˆp′(s)|≤12nf(n)]≥1−1exp(n).

We compute for all with , and define (as described at the end of Section 3.2) and for all with . Since , it takes polynomial time to compute all these values. Moreover,

 Pr[∀s∈{0,1}n with |s|≤c, |ˆp(s)−ˆp′(s)|≤12nf(n)]≥1−1exp(n).

This can be simply shown by a direct application of the inequality for an arbitrary real number and integer .

We define a function over as

 q(x)=∑s∈{0,1}n,|s|≤c(1−λ)|s|ˆp′(s)(−1)s⋅x.

In the following, we show that under the assumption that, for all with , . A direct calculation with the relations described in Section 2.2 shows that

 ||˜pA−q||21 ≤2n||˜pA−q||22=22n||ˆ˜pA−ˆq||22 =22n∑s:|s|>c(1−ε)2|s|ˆp(s)2+22n∑s:|s|≤c[(1−ε)|s|ˆp(s)−(1−λ)|s|ˆp′(s)]2. (2)

Using the bounds and , we upper-bound the first term of (4.1) as follows:

 22n∑s:|s|>c(1−ε)2|s|ˆp(s)2 ≤22n(1−λ)2c∑s:|s|>cˆp(s)2≤22n(1−λ)2c∑s∈{0,1}nˆp(s)2 =2n(1−λ)2c∑x∈{0,1}np(x)2≤2ne2λc⋅α2n≤α22λc≤δ2100.

Then, we upper-bound the second term of (4.1) as follows:

 22n∑s:|s|≤c[(1−ε)|s|ˆp(s)−(1−λ)|s|ˆp′(s)]2 =22n∑s:|s|≤c{[(1−ε)|s|−(1−λ)|s|]ˆp(s)+(1−λ)|s|(ˆp(s)−ˆp′(s))}2 ≤22n∑s:|s|≤c{[(1−λ)|s|−(1−ε)|s|]|ˆp(s)|+(1−λ)|s||ˆp(s)−ˆp′(s)|}2 ≤22n∑s:|s|≤c(c(ε−λ)|ˆp(s)|+12nf(n))2, (3)

where the last inequality is due to the fact that and , where .

Using the bounds , ,