Bounds on stabilizer measurement circuits and obstructions to local implementations of quantum LDPC codes

09/29/2021
by   Nicolas Delfosse, et al.
0

In this work we establish lower bounds on the size of Clifford circuits that measure a family of commuting Pauli operators. Our bounds depend on the interplay between a pair of graphs: the Tanner graph of the set of measured Pauli operators, and the connectivity graph which represents the qubit connections required to implement the circuit. For local-expander quantum codes, which are promising for low-overhead quantum error correction, we prove that any syndrome extraction circuit implemented with local Clifford gates in a 2D square patch of N qubits has depth at least Ω(n/√(N)) where n is the code length. Then, we propose two families of quantum circuits saturating this bound. First, we construct 2D local syndrome extraction circuits for quantum LDPC codes with bounded depth using only O(n^2) ancilla qubits. Second, we design a family of 2D local syndrome extraction circuits for hypergraph product codes using O(n) ancilla qubits with depth O(√(n)). Finally, we use circuit noise simulations to compare the performance of a family of hypergraph product codes using this last family of 2D syndrome extraction circuits with a syndrome extraction circuit implemented with fully connected qubits. While there is a threshold of about 10^-3 for a fully connected implementation, we observe no threshold for the 2D local implementation despite simulating error rates of as low as 10^-6. This suggests that quantum LDPC codes are impractical with 2D local quantum hardware. We believe that our proof technique is of independent interest and could find other applications. Our bounds on circuit sizes are derived from a lower bound on the amount of correlations between two subsets of qubits of the circuit and an upper bound on the amount of correlations introduced by each circuit gate, which together provide a lower bound on the circuit size.

READ FULL TEXT VIEW PDF

Authors

page 1

page 2

page 3

page 4

06/20/2019

Exponential separation between shallow quantum circuits and unbounded fan-in shallow classical circuits

Recently, Bravyi, Gosset, and König (Science, 2018) exhibited a search p...
11/03/2020

Circuit lower bounds for low-energy states of quantum code Hamiltonians

The No Low-energy Trivial States (NLTS) conjecture of Freedman and Hasti...
07/13/2020

In-place implementation of Quantum-Gimli

We present an in-place implementation of the Gimli permutation, a NIST r...
06/30/2019

Logical Clifford Synthesis for Stabilizer Codes

Quantum error-correcting codes are used to protect qubits involved in qu...
09/29/2021

Constant-overhead quantum error correction with thin planar connectivity

Quantum LDPC codes may provide a path to build low-overhead fault-tolera...
12/03/2020

Quantum learning algorithms imply circuit lower bounds

We establish the first general connection between the design of quantum ...
01/31/2022

A lower bound on the space overhead of fault-tolerant quantum computation

The threshold theorem is a fundamental result in the theory of fault-tol...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

1 Introduction and overview

Quantum Low Density Parity Check (LDPC) codes [25, 34, 18, 29], capable of encoding many logical qubits within the same block, offer promising performance for large scale fault-tolerant quantum computation. Quantum LDPC codes have two appealing features: (i) they are defined by constraints involving a small number of qubits, (ii) some can encode logical qubits into physical qubits with , with a large minimum distance . Recall that a minimum distance guarantees that the code can correct any error acting on up to qubits111In the ideal case of perfect syndrome extraction.. Property (i) makes quantum LDPC codes easier to implement than general codes, and (ii) implies quantum LDPC codes perform well and could lead to a more favorable overhead than the surface code [12, 9].

Numerical simulations of the performance of quantum LDPC codes without geometric locality constraints show encouraging results [20, 14, 15, 27]. In this work, we explore the potential of quantum LDPC codes implemented with quantum hardware limited to local operations in a 2D grid of qubits which matches the capability of many hardware technologies such as superconducting qubits [2, 6] and Majorana qubits [19].

To correct errors affecting an encoded quantum state, we first execute a quantum circuit, the so-called syndrome extraction circuit, which measures the local constraints defining the code. Then, the syndrome output by the syndrome extraction circuit is fed to the decoder which is a classical subroutine that processes the syndrome and returns a correction. Finally, the encoded state is corrected using the output of the decoder.

Decoders have been proposed for quantum LDPC codes [22, 28, 31, 8] which can be combined with any syndrome extraction circuit. With 2D connectivity, we expect the bottleneck for error correction performance to be the syndrome extraction circuit. However, the question of designing 2D local syndrome extraction circuits for quantum LDPC codes remains open.

Locality is known to restrain the parameters achievable with quantum error correction codes. Bravyi, Poulin and Terhal [4] proved that quantum codes defined by local commuting projectors in a 2D grid of qubits obey the bound . This result translates into a bound on the depth of syndrome extraction circuits built from with local unitary gates and local measurements in a 2D square patch. Indeed, assume that a family of quantum codes with length encoding

logical qubits can be implemented with 2D local syndrome extraction circuits with bounded depth. The projectors defining a code can be obtained by backpropagating the projectors corresponding to local measurements in the circuit. For bounded-depth local syndrome extraction circuits, this procedure leads to a family of local commuting projectors in 2D. If the circuit is supported on a square patch of

qubits, this implies that the parameters of the codes are limited by the tradeoff which leads to . The only quantum codes that can be implemented in bounded depth with these assumptions have a bounded minimum distance and therefore a non-vanishing logical error rate (and no threshold).

Figure 1: A local circuit on a line for qubits implementing the measurement of the Pauli operator supported on the endpoints of the line (qubit 1 and qubit 7). Each qubit comes with a classical memory represented by the light blue line which is used to store the measurement outcomes. The circuit is made with single-qubit and two-qubit quantum operations, represented in black, acting on adjacent qubits. The blue boxes represent classically-controlled Pauli operations that are applied only if a subset of bits has parity . These conditional gates may depend on any subset of bits and require long-range classical communication, represented by blue vertical dashed lines to access the classical bit stored.

This argument does not preclude bounded-depth syndrome extraction circuits for general quantum LDPC codes with realistic hardware constraints in 2D because it does not allow for classical communication. It is reasonable to assume that a quantum chip is equipped with at least some long-range classical communication channels. As an example, the circuit of Fig. 1 can be used to perform a Pauli measurement between the two endpoints of a line of qubits in bounded depth using long-range classical communication. However, it is unclear if we can simultaneously measure a large set of Pauli operators in bounded depth.

Moreover, if the syndrome extraction cannot be implemented in bounded depth using a total of qubits, then what is the minimum depth of a 2D local syndrome extraction circuit using qubits? Alternatively, what is the minimum number of ancilla qubits necessary to implement the syndrome extraction in bounded depth with a 2D local circuit? Finally, how do quantum LDPC codes perform with these syndrome extraction circuits?

We explore all these questions in this article. We consider a class of 2D local Clifford circuits made with single-qubit and two-qubit unitary Clifford gates and single-qubit and two-qubit Pauli measurements. We also authorize classically-controlled Pauli operations controlled on the parity of any set of outcomes of previous measurements. By extending Clifford operations with these classical controlled gates, which exploit long-range classical communication, we obtain a class of Clifford circuit which can implement non-local Pauli measurement in bounded depth through cat states. We assume no restriction on the classical storage of the measurement outcomes or the classical communication. As a result, the parities controlling different conditional Pauli operations of the same layer of a circuit can be computed instantaneously and simultaneously even if some conditions depend on the same bits or depend on bits that are stored far away from the qubits supporting the Pauli operation.

Our main technical result is a general lower bound on the depth of Clifford circuits implementing the measurement of a family of Pauli operators. This bound relies on the connectivity graph of the circuit which is the graph whose vertices support the qubits and such that two qubits are connected if the circuit contains a two-qubit operation acting on them.

Theorem 1.

Let be a Clifford circuit measuring commuting Pauli operators . Then, for any subset of qubits , we have

where is the number of independent operators with support on both and its complement and is the set of edges connecting and its complement in the connectivity graph of the circuit.

This result is obtained by studying the correlations created between two subsets of qubits of the circuit. We believe that our proof technique could find other applications to place bounds on quantum circuits and we provide an overview of our proof strategy in Section 4.1.

Applying Theorem 1, we obtain bounds on the circuit depth and the number of ancilla qubits required to perform the syndrome extraction of quantum LDPC codes in different settings. For families of local-expander quantum LDPC codes with length , if the syndrome extraction is implemented with a 2D local Clifford circuit acting a patch of qubits, we prove (Corollary 1) that the depth of the syndrome extraction circuits satisfies

(1)

We establish similar results for -dimensional local Clifford circuits (Corollary 2) and for 2D local Clifford circuits acting on an arbitrary subset of the square grid (Corollary 3). The assumption of local-expansion is necessary because surface codes violate these bounds. However, their parameters are limited by the tradeoff [4]. Many standard families of classical and quantum LDPC codes exhibit local-expansion such as random classical LDPC codes[11], classical expander codes [32], quantum hyperbolic codes [35, 16] and quantum hypergraph product codes [34]. The assumption of local expansion is also commonly used to guarantee that the decoder performs well [32, 17, 22, 8] and some notion of expansion is known to be required to obtain good quantum LDPC codes [3].

Then, we propose two classes of syndrome extraction circuits saturating the bound (1). On the one hand, we design 2D local Clifford syndrome extraction circuits for any family of CSS codes that run in bounded depth and use only ancilla qubits which is the minimum number of ancilla qubits for a bounded depth circuits based on (1). On the other hand, we design 2D local Clifford syndrome extraction circuits for any family of hypergraph product codes [34] that run in depth and use only ancilla qubits. Based on the bound (1), this is the best achievable depth when using ancilla qubits.

Figure 2: Logical failure rate as a function of the gate error rate for a family of hypergraph product codes with rate 1/25, using either the 2D local syndrome extraction circuit described in Section 6 or a bounded-depth syndrome extraction circuit described in the beginning of Section 5 made with two-qubit gates acting on arbitrary pairs of qubits.

To assess the performance of quantum LDPC codes of finite (rather than asymptotic) length, we perform simulations with good 2D local syndrome extraction circuits. In particular, we simulate the performance of a family of hypergraph product codes [34] with encoding rate . To preserve the high rate of logical qubits per physical qubits of these codes, we use our second family of syndrome extraction circuits which uses ancilla qubits. Including the ancilla qubits used for syndrome extraction, the rate of the family is . Fig. 2 shows the performance of this family of quantum LDPC codes in two different settings. The code and the decoder are the same in both settings but the syndrome extraction is implemented either with our 2D local syndrome extraction circuit or with a bounded-depth syndrome extraction circuit using arbitrary two-qubit gates, ignoring geometric constraints. This second case assumes fully connected qubits. Both schemes are simulated with circuit level noise. In the regime of our simulation, we observe that the 2D local implementation of this family of quantum LDPC codes achieves a performance comparable to the fully connected implementation with a physical noise rate increased by a factor 100. Moreover, we observe in Fig. 2 that increasing the block size degrades the performance in 2D local case which shows that if this family of codes has a non-zero error threshold in 2D, it is below . In a companion paper [27], we explore an alternative implementation of these quantum LDPC codes using some long-range connections between the qubits.

We introduce definitions and background material in Section 2. Our bounds on the depth and the number of ancilla qubits of local syndrome extraction circuits are proven in Section 3. The proof of our main technical result, Theorem 1, is provided in Section 4. Section 5 describes a construction of bounded-depth 2D local syndrome extraction circuits for CSS codes. The 2D local syndrome extraction circuits for hypergraph product codes using a linear number of ancilla qubits are proposed in Section 6. The last section, Section 7, describes our numerical results.

2 Background and definitions

Here we review some background topics and provide definitions that will be used throughout the rest of the paper.

2.1 Clifford circuits

A quantum circuit is a sequence of operations acting on a set of qubits. There are many important classes of circuit, which can typically be defined by restricting the set of allowed operations from which the circuit is composed. In this work, we focus on Clifford circuits which are built from the following operations.

  • Preparations of or .

  • Single-qubit and two-qubit Pauli measurements.

  • Single-qubit and two-qubit unitary Clifford gates.

  • Classically-controlled Pauli operations, applied only if some subset of previous measurement outcomes has parity 1.

  • Output a set of classical bits obtained by computing the parity of some subsets of measurement outcomes.

To avoid any confusion with the outcomes of the single-qubit and two-qubit measurements of the circuit, we reserve the term output to refer to the parity bits returned at the end of the circuit. We assume that the Pauli operation has access to all the measurement outcomes extracted in the past. We discuss the generalization of our results to circuits involving more general operations in Section 3.4.

A Clifford circuit is decomposed into layers of the above operations in such a way that the support of two operations of the same layer do not overlap and the depth of the circuit is the number of layers. Here we define the support of a classically-controlled Pauli operation to be the set of qubits acted on by the Pauli operation. The support of a classically-controlled Pauli operation is independent of the classical bits controlling the gate. Therefore, the conditions of two classically-controlled Pauli operations of the same layer may overlap. We assume unlimited classical communication and classical storage capacity and do not account for any delay due to classical control. For example, Fig. 1 shows a circuit with depth six. Here, we include Pauli operations in the calculation of the depth of the circuit. Given that Pauli operations can be implemented by frame update, which requires only classical processing, we could also ignore Pauli operations in the calculation of the depth of quantum circuits. With this notion of depth, our lower bounds on the depth of a quantum circuit would remain unchanged up to a constant factor.

Given a circuit , we define the connectivity graph of the circuit which has a vertex for each qubit, and an edge connecting each pair of vertices that are in the support of any single operation applied in the circuit.

On the other hand, consider a set of qubits whose connectivity is described by a graph , i.e. the two-qubit gates available are the gates supported on the edges of the graph . A circuit is said to be implementable with qubits with connectivity if is a subgraph of .

A -dimensional -local circuit is a circuit acting on qubits placed on a subset of the grid such that each gate acts on qubits at distance at most from each other, where distances are calculated with the infinity norm. We sometimes refer to a family of -dimensional -local circuits as a family of -dimensional local circuits, omitting the constant . For example, measuring the stabilizer generators of the surface code can be implemented by a two-dimensional -local circuit family of bounded depth using a square-grid connectivity graph [10].

2.2 Pauli measurement circuits

Consider a set of commuting -qubit independent Pauli operators . For , denote by

the projector onto the common eigenspace of the

with eigenvalue

. Following the postulates of quantum mechanics, a Pauli measurement circuit for the measurement of is a Clifford circuit which takes as an input a -qubit state with density matrix and returns the output

with probability

Moreover, the state of the input qubits must be mapped onto

We call the qubits supporting the operators in the data qubits, and any number of other qubits in the circuit the ancilla qubits such that the total number of qubits in the circuit is . We assume that ancilla qubits are prepared in the state or before being used and they are returned to a state or at the end of the circuit. We also assume that data qubits are never measured with a single qubit measurement.

Like in Section 2.1, we assume that the output , corresponding to the measurement of , is obtained by taking the parity of a subset of outcomes of single-qubit and two qubit measurements of the circuit, that is where denotes the sum modulo 2. These outcomes may be extracted from any layer of the circuit and the sets may overlap.

2.3 Tanner graph and contracted Tanner graph

Consider a set of -qubit Pauli operators acting on qubits that we denote . The Tanner graph of the set of Pauli operators , denoted or simply when no confusion is possible, is defined to be the bipartite graph where and correspond respectively to the data qubits and the Pauli operators. The vertices and are connected by an edge if and only if the operator acts non-trivially on qubit .

The contracted Tanner graph or simply , is the graph with vertex set such that two vertices and are connected by an edge if and only if there exists an operator that acts non-trivially on both and . Alternatively, the contracted Tanner graph can be obtained by connecting two data qubits if and only if they are at distance two in the Tanner graph.

2.4 Stabilizer codes and quantum LDPC codes

A stabilizer code with length is the common -eigenspace of a set of -qubit independent commuting Pauli operators . We refer to the operators as the stabilizer generators of the code. When considering a stabilizer code, we always assume that it comes with fixed set of stabilizer generators. By the Tanner graph of a stabilizer code, we mean the Tanner graph of its set of stabilizer generators.

By a syndrome extraction circuit for the stabilizer code with stabilizer generators , we mean a Pauli measurement circuit for the set .

A family of quantum LDPC codes is a family of stabilizer codes such that the Tanner graph of has degree bounded by some constant independent of . Equivalently, there exists a constant such that for all , each stabilizer generator has weight at most and each qubit is acted on by at most stabilizer generators.

In Sections 5 and 6, we focus on Calderbank Shor Steane (CSS) codes [5, 33]. These are codes for which each stabilizer generator is either composed entirely of and operators or entirely of and operators, such that where is the set of -type stabilizer generators etc. In the case of CSS codes, we define the Tanner graph as the subgraph of induced by the vertices corresponding to the qubits and the stabilizers. We define the Tanner graph similarly.

2.5 Expander graphs and expander codes

We introduce the following local generalization of the Cheeger constant of a graph , denoted , defined as

for all . Therein, is the boundary of , that is the set of edges connecting and its complement. The usual Cheeger constant, denoted is obtained by setting , that is .

A family of -expander graphs is a family of graphs such that for all . We consider a generalization of this notion by considering expansion over small subsets of vertices. A family of -expander graphs is a family of graphs such that for all .

Clearly, if is a family of -expander graphs, it is also a family of -expander graphs for all . However, -expansion with does not imply -expansion.

By a family of quantum expander codes we mean a family of quantum LDPC codes such that the family of contracted Tanner graphs of the stabilizer generators is a family of -expander graphs for some . Similarly, we define a family of local-expander codes as a family of quantum LDPC codes equipped with -expander contracted Tanner graphs for some . By definition, a family of expander codes is also a family of local-expander codes.

It is common in the literature to define quantum expander codes or local quantum expander codes in terms of their Tanner graphs rather than their contracted Tanner graphs. In Lemma 1, we show that if a code’s Tanner graph is a local expander graph, then so too is its contracted Tanner graph. This allows our results to be applied to code families with known expansion properties of the Tanner graphs.

Lemma 1.

Let be the Tanner graph of a stabilizer code with length and with stabilizer generators and let be its contracted Tanner graph. Then, for all , we have

where .

Proof.

A subset can be treated as a subset of . To avoid any confusion, we use the notation and for the set of edges leaving in the two graphs.

To enumerate edges of , we introduce a function where denotes the power set of . The function maps the edge with and onto the set of edges of such that is adjacent to both and in . On the one hand, there are at most edges in the image of an edge and each edge of belongs to one of the subsets of the image of . This shows that . On the other hand, we have . This proves that

(2)

Let and let . Consider a subset of such that . Then, we have

(3)
(4)
(5)

Therein, we used and .

Given that , we can use the Cheeger constant of , which yields

(6)

Combining this with Eq. (2), we get

(7)
(8)

proving the lemma. ∎

3 Circuit bounds

Our main technical result (Theorem 1) establishes a lower bound on the depth of Clifford circuits measuring a set of commuting Pauli operators . In what follows, denotes the contracted Tanner graph of the set of measured operators . In this section, we provide different applications of Theorem 1. The proof of this theorem is deferred to Section 4.

3.1 Local circuits on qubits on a patch of

In what follows, we use the terminology of an -qubit patch of to refer to a square grid of qubits.

Proposition 1.

Let be a 2D -local Clifford Pauli measurement circuit supported on an -qubit patch of . Then, for all we have

for some constant , where and are respectively the contracted Tanner graph and maximum weight of the set of measured Pauli operators.

Proof.

Consider the patch with integer lattice coordinates . For an integer , the subset of qubits supported on the vertical line is denoted and is the union of all the sets with .

Let be the set of data qubits of the circuit. By shifting , we can find a position such that

This value exists because when moves by one unit at most data qubits change side.

Denote by the number of vertices contained in a closed ball with radius in . Using the -locality of the circuit, we see that the set of all the qubits included in satisfies

(9)

for some constant .

Moreover, using the Cheeger constant of the contracted Tanner graph, the number of operators with support on both and its complement is at least

(10)
(11)

The term is present because each induces at most edges in the contracted Tanner graph.

Applying Theorem 1 with this set , we get

(12)

for some constant . ∎

The following corollary is a straightforward application of Proposition 1.

Corollary 1.

Let be a family of 2D local Clifford syndrome extraction circuits for a family of local-expander quantum LDPC codes with length where acts on qubits in a patch of . Then, we have

where is the total number of qubits used by the circuit.

For simplicity, we have considered here a circuit with qubits filling a square grid. However Corollary 1 also holds for a family of circuits acting on a subset of qubits occupying a constant fraction of this square grid because this is equivalent to replacing by . Therefore, this corollary holds for qubits placed on a hexagonal or a triangular lattice. The proof of Proposition 1 and the statement of Corollary 1 can be readily modified to apply to rectangular patches , where and can grow at different paces in the code family.

3.2 Local circuits on qubits on a patch of

By an -qubit patch of we mean a -dimensional cubic grid of qubits where the length of each side of a -cube is . Proposition 1 immediately generalizes to -dimensions as follows.

Proposition 2.

Let . Let be a -dimensional -local Clifford Pauli measurement circuit acting on an -qubit patch of . Then, for all we have

for some constant , where and are respectively the contracted Tanner graph and maximum weight of the set of measured Pauli operators.

We now state the -dimensional analog of Corollary 1 which is an immediate application of this proposition.

Corollary 2.

Let . Let be a family of -dimensional local Clifford syndrome extraction circuits where acts on qubits in a patch of for a family of local-expander quantum LDPC codes with length . Then, we have

where is the total number of qubits used by the circuit.

3.3 Local circuits on qubits on any subset of

Here we consider circuits supported on an arbitrary subset of the square grid .

Proposition 3.

Let be a 2D -local Clifford Pauli measurement circuit using a total of qubits on any subset of . Then, for all we have

for some constant , where and are respectively the contracted Tanner graph and maximum weight of the set of measured Pauli operators.

Proof.

To construct the subset of Proposition 1, we use a separation theorem [23]. We use the variant of the separation theorem proposed in Theorem 3 of [24] which provides a decomposition of a planar graph into connected components with size by removing at most vertices.

Denote by the size of a closed ball with radius . Recall that we use the infinity distance. We apply the aforementioned separation theorem with , to the subgraph of the square grid containing all the nodes at distance less or equal to from any of the sites which contain a qubit. Let be the vertex sets of the subgraphs of the decomposition and let be the set of removed vertices. For , we have

(13)

because the graph contains at most vertices.

Denote and let be the set of data qubits of the circuit included in . Select the minimum index such that . Then, we have

(14)

The upper bound is clear by the definition of . The lower bound is due to the fact that discarding removes at most data qubits based on Eq. (13).

Let be the set of all the qubits included in . Let us derive an upper bound on the size of in the connectivity graph. If with and , there exists a path with length joining and in the square grid and this path contains at least one vertex of the removed set . This induces a map from to and each vertex of has at most preimages because and are at distance at most from . Therefore, we have

(15)
(16)
(17)
(18)

for some constant .

Moreover, like in the proof of Proposition 1, using the Cheeger constant of the contracted Tanner graph, we get

(19)

We conclude by applying Theorem 1 with this set using Eq. (18) and Eq. (19) which yields

(20)

Applying this result to syndrome extraction circuits for families of local-expander quantum LDPC codes, we obtain the following result.

Corollary 3.

Let be a family of 2D local Clifford syndrome extraction circuits for a family of local-expander quantum LDPC codes with length acting on qubits on any subset of . Then, we have

where is the total number of qubits used by the circuit.

The bound obtained in this section for local circuits on a general subset of is weaker than the bound obtained for local circuits on a patch of in Proposition 1. We do not know if this particular bound is tight – i.e. if there is a smaller-depth syndrome using 2D local extraction circuit acting on qubits placed on a subset of . However, we note that bounded-depth circuits using a linear number of ancilla qubits are still excluded.

3.4 Generalizations

Our results can be immediately generalized to other classes of quantum circuits.

First, above we assumed that the Pauli operators measured in a Pauli measurement circuit are independent, but our results directly apply to the measurement of non-independent operators by considering an independent subset.

For simplicity, we assume that circuits involve operations acting on at most two qubits. We can generalize our results to Clifford circuits including Clifford gates and Pauli measurements supported on up to qubits. In this case, the connectivity graph of the circuit is a hypergraph defined in such a way that each gate is supported on a hyperedge and the bound of Theorem 1 becomes

where . Therein, denotes the set of hyperedges containing at least one vertex in each set and . The value of the constant corresponds to the maximum increase of entanglement entropy induced by a -qubit unitary gate (Proposition 2 in [26]) which is central in the proof of Proposition 4.

The results of this article also hold for some non-Clifford circuits. Using the exact same proof technique, one can extend our results to the case of quantum circuits made with preparations of or , unitary gates acting on up to qubits, and single-qubit measurements in the computational basis. However, in this class of circuits, we do not allow classical communication and classically-controlled Pauli operations. We leave the case of unitary circuits with unconstrained classical communication open.

4 Proof of Theorem 1

This section provides the proof of Theorem 1.

4.1 Proof strategy

In this subsection, we sketch the overall strategy for our proof before providing and proving the formal Lemmas which comprise the rigorous proof in the following sections.

We consider a Pauli measurement circuit which returns the outcomes of the measurement of a set of Pauli operators . Our strategy to bound the depth of is to study correlations introduced by the circuit across a partition of the qubits into two subsets, and . On the one hand, because the circuit implements a non-trivial operation, the two sides of the partition are generally correlated at the end of the circuit. We can use this to derive a lower bound on the amount of correlation created by the circuit. On the other hand, building these correlations with local gates takes time because we expect that each gate introduces a bounded amount of correlation. We can use this to derive an upper bound on the amount of correlation created by a circuit as a function of its depth. Combining both arguments, we obtain a lower bound on the depth of the circuit.

To apply this strategy, we need a protocol which uses the circuit and a measure of correlation that allows us to readily quantify correlations to build both the lower bound and the upper bound. In what follows we consider a number of relevant aspects which lead us to such a protocol and measure.

Initial state. To avoid the presence of correlation in the initial state of the system, we fix the input state to be for the data qubits.

Measure of correlation. Different notions can be used to capture correlations between the two parts of the partition such as the classical mutual information, the entanglement entropy or the quantum mutual information. Our starting point is to study the classical mutual information between the measurement outcomes extracted in each side of the partition. We will refine this notion throughout this section.

Repeated circuit. Consider as an example the Pauli measurement circuit of Fig. 1 which measures the operator supported on the endpoints of a line of qubits, with all qubits initially in the state. Pick the partition of the qubits where contains the first four qubits and contains the three remaining qubits. The output of the circuit is . Since the input state is fixed to , the value of will be a uniform random bit, and there is therefore no correlation between the outcomes in and . However, if we run the measurement circuit a second time, we obtain the same output , even though each individual measurement outcome is a uniform random bit independent of the previous outcomes . Then, because the circuit output is fixed to , there is one bit of classical mutual information between the outcomes observed on each side during the second run, that is

This example encourages us to consider the circuit and to use the notion of classical mutual information between measurement outcomes across the two sides of the the partition to capture correlations

where (respectively ) denotes the set of all measurement outcomes extracted on the qubits of (respectively ) during the second run of .

Highlighting correlations with errors: The example considered above allowed us to track the build up of correlations because the output of the simple measurement circuit depends on some measurement outcomes extracted on each side of the partition. However, if a Pauli operator is measured using a single ancilla qubit, or a set of ancillas that are all supported on the same side of the partition, the argument breaks down. To highlight the correlations present in the system and that cannot be detected by looking only at the outcomes, we introduce a layer consisting of a random Pauli error acting on the data qubits between the two runs of and we consider the circuit . If a measured operator is supported on both sides of the partition, a Pauli error acting on one side may flip the outcome observed on the other side. Therefore, we measure the correlations using the mutual information

where and are the restrictions of the Pauli error to each subset of qubits.

Discounting correlations due to classical communication: We have argued that we can derive a lower bound on the mutual information between the two sides of a partition of the qubits based on the stabilizer generators which are supported on both sides of the partition. To apply our strategy we also need an upper bound on the amount of correlation as a function of the circuit depth. Unfortunately, the quantity is not an appropriate measure of correlation to derive a non-trivial upper bound. This is because the Clifford circuits that we consider include classical communication through classically-controlled Pauli operations and these operations can boost the mutual information as the illustrated by the following example.

One can create bits of mutual information in bounded depth for a partition of qubits using only single qubit operations and classical communication as follows. Group qubits into pairs, with each pair containing one qubit of in the state and one qubit of in the state . First, we measure all the states in the computational basis and if the outcome is non-trivial we apply an gate to the other qubit of the pair. Then, we measure all the qubits in the computational basis. This constant-depth circuit produces one bit of mutual information for each pair of qubits. This proves that, using classical mutual information as a measure of correlation, we would not be able to derive a sufficiently good upper bound on the amount of correlations as a function of the circuit depth. The entanglement entropy, which is a standard measure of correlation in a bipartite quantum system suffers from the same issue.

To avoid this issue of classical communication introducing mutual information we first transform the circuit into a circuit (where is guaranteed to have the same action and a similar depth to ) by postponing the measurements and the classical controlled Pauli operations until the end of the circuit. Moreover, to avoid the increase of mutual information by the controlled-Pauli operations at the end of the first run of , we use the conditional mutual information

conditioned on the outcomes of the first run of .

This notion of correlation is adapted to obtain both a non-trivial lower bound and a non-trivial upper bound on the amount of correlation between two subsets of qubits and and leads to the bound of Theorem 1.

The proof of Theorem 1 relies on Lemma 4 and Lemma 5 proven in the following subsections which provide a lower bound and an upper bound on the conditional mutual information . Instead of using the circuit , we use the version of this circuit obtained through the circuit transformations described in Lemma 2 and Lemma 3.

4.2 Circuit transformation

Instead of working with the Clifford circuit , we build a modified version of which implements the same Pauli measurements on the data qubits but which is easier to analyze.

Figure 3: Transformed Clifford circuit obtained in Lemma 2. Any Clifford circuit can be transformed into such a four-stage circuit made with layers of unitary Clifford gates, followed by a layer of single-qubit measurements and two layers of classically-controlled Pauli operations. In this figure, each wire represents a block of multiple qubits, is a product a single-qubit and two-qubit Pauli operations, the blocks and represent a layer of classically-controlled single-qubit Paulis.
Lemma 2.

Let be a Clifford circuit implementing an operation on a set of data qubits D using a set of ancilla qubits A. Then, there exists a circuit represented in Fig. 3 which implements the same operation as on D, using a set of ancilla qubits , made with the following steps.

  1. The preparation of all the ancilla qubits in is in the state or .

  2. A unitary Clifford circuit with built from single-qubit and two-qubit unitary Clifford gates.

  3. A layer of single-qubit measurements of every qubit in with outcome set , followed by classically-controlled and on depending on .

Moreover, any subset of qubits of maps onto a subset such that each layer of the circuit contains at most gates supported on both and its complement.

Therein, the notation refers to boundary edges relative to the connectivity graph of the circuit .

Proof.

We start with the circuit and carry out a sequence of modifications which do not change the action of the circuit until we reach . First we eliminate two-qubit measurements. Each joint measurement is replaced by a pair of two-qubit gates and a single-qubit measurements using the identify in Fig. 4. This transformation increases the circuit depth by at most a factor four.

Given , we initially define . If the joint measurement is supported on two qubits of (respectively ), the new ancilla qubit is added to (respectively ). If the joint measurement is supported on an edge of , then we add the ancilla qubit to . By definition of , each layer of the circuit after this transformation contains at most gates which act non-trivially on both and its complement.

Figure 4: Simulation of the measurement of a two-qubit Pauli operator using unitary Clifford gates, single-qubit measurements and an ancilla qubit.

Then, each ancilla qubit is replaced by a set of ancilla qubits where is the number of times is measured in the original circuit. These ancilla qubits are all initialized in and at most one of them can be involved in a non-trivial operation in a given time step. Let be the time steps where is measured and denote . The ancilla qubit plays the role of the qubit in the original circuit during all the time steps from to (included). This transformation guarantees that each ancilla qubit is measured exactly once. If the original ancilla qubit is in (respectively ), we assign all its copies to (respectively ).

The number of gates supported on both and its complement within a layer of the circuit is unchanged during this transformation. This is because only one of the copies of an ancilla qubit is used at a given time step.

Figure 5: Effect of the commutation of a Pauli measurement and a classically-controlled Pauli operation. If the Pauli operation anti-commutes with the measured operator, swapping them introduces a XOR of the measurement outcome with the condition.
Figure 6: Effect of the XOR of a stored bit on a circuit output.
Figure 7: Up to a global phase, any sequence of consecutive classically-controlled Pauli operators acting on a single qubit can be decomposed into a classically-controlled followed by a classically-controlled . The conditions is and where is the set of indices such that or and is the set of indices such that or .

Now, we move all the classically-controlled Pauli operations to the end of the circuit. One can move a Pauli operation passed a Clifford gate using the relation = where is also a Pauli operation. To move a classically-controlled Pauli operation passed a measurement, one can use the relations of Fig. 5. Then, we combine the sequences of classically-controlled Paulis into a layer of classically-controlled and a layer of classically-controlled using the identity of Fig. 7. The classically-controlled Pauli operations at the end of the new circuit after the ancillas have been measured can be decomposed into those acting entirely on data qubits and those acting entirely on ancilla qubits. We discard those acting on the ancillas because they only change the parities used to produce the circuit outputs as explained in Fig. 6. These moves and other circuit transformations keep invariant.

Finally, we can trivially postpone the measurements to perform them after the unitary gates and before the classically-controlled Pauli operations because ancilla qubits are not reused after measurement in this transformed circuit. We can also implement all the measurements simultaneously for the same reason. Again, the set is kept unchanged. ∎

4.3 The double measurement circuit

In this section, we consider the circuit which runs a circuit which measures Pauli operators followed by by a uniformly drawn random Pauli error on the data qubits before running again. In the following lemma, we form a simplified version of the circuit using the transformation of Lemma 2.

Lemma 3.

Let be a Pauli measurement circuit on a set of data qubits D using a set of ancilla qubits A. Then, there exists a circuit represented in Fig. 8 which implements the same operation as on D, using a set of ancilla qubits , made with the following steps.

  1. The preparation of all the ancilla qubits in in the state and all the other ancilla in either the state or .

  2. A unitary operation on with made with single-qubit and two-qubit unitary Clifford gates.

  3. A uniform random Pauli error generated by measuring the ancilla qubits of in the computational basis and applying conditional Pauli error based on these measurement outcomes.

  4. The same unitary operation applied to .

  5. A layer of single-qubit measurements of all the qubits in with outcome set , followed by classically-controlled and on depending on .

  6. A layer of single-qubit measurements of all the qubits in with outcome set , followed by classically-controlled and on depending on .

Moreover, any subset of qubits of maps onto a subset such that each layer of the circuit contains at most gates supported on both and its complement.

Like in Lemma 2, the notation refers to boundary edges relative to the connectivity graph of the circuit .

Figure 8: The circuit obtained from by applying Lemma 2 to each of the two rounds of . In this figure, each wire represents a block of multiple qubits, is a product a single-qubit and two-qubit Pauli operations, the blue dashed blocks represent layers of classically-controlled single-qubit Paulis.
Proof.

Starting from the circuit , we build the circuit by applying the transformation of Lemma 2 to . Each round uses ancilla qubits. We assume that the ancilla qubits are not reused and we denote by and the set of ancilla qubits used by the first and the second application of . The circuit includes a random Pauli error acting on the data qubits which we generate with classically-controlled Pauli operations which depend on the outcomes of the measurement of states in the computational basis. Each data qubit corresponds to a pair of states . We apply an gate (respectively a gate) to the qubit controlled on the value of the outcome of the measurement of (respectively ).

Finally, we move the first two rounds of classically-controlled Pauli operations associated with the application of the first measurement circuit to after the second block of unitary gates . This is done by conjugating and by and separating the part and the part as using the relation of Fig. 7, which results in the Pauli operations and in Fig. 8.

Given , the application of Lemma 2 to each copy of maps onto a subset of and it guarantees that each layer contains at most gates acting non-trivially on both and its complement. The set is obtained from by adding the ancilla qubits of used to generate errors on the data qubits of . Moving the classically-controlled gates cannot introduce any operation acting on and its complement because these gates are decomposed into single-qubit Pauli operations. ∎

4.4 Notation

In what follows, we consider a Pauli measurement circuit for the measurement of a set of Pauli operators . The circuit is the circuit obtained in Lemma 3 by simplifying the circuit where is a round of Pauli errors. We refer to the circuit , represented in Fig. 8, as the double measurement circuit.

Denote by or the sets of outcomes extracted during the two runs of . The circuit outputs the values where and . The output is the outcome of the measurement of the Pauli operators  produced by the th run of . It is obtained as

for some subset of .

In what follows, we consider a partition of the qubit set as with . The subset of the qubit set of the double measurement circuit induced by is denoted and its complement is denoted . Throughout the proof, when we use the notation , it is always for a set induced by a subset of .

The sets of outcomes obtained during the first and the second application of split along the partition as . We use the notation and for the restrictions of the Pauli error to and respectively.

4.5 Lower bound on the mutual information

Here, we derive a lower bound on the mutual information between the outcomes of the double measurement circuit obtained on each side of a partition of the qubits into two subsets. We use the notations of Fig. 8 and Section 4.4.

Lemma 4.

With the notations of Section 4.4, we have

where is the number of operators supported on both and .

Proof.

By the data processing inequality, we have

(21)
(22)

where and are the parities corresponding to the restrictions of the sets to each subset and . For instance, is the set of values . Then, using the relation , we obtain

(23)
(24)
(25)

Consider the term in Eq. (25). Let us show that the values in are fully determined by and . We have where is if and commute and otherwise and is defined similarly for the product of the conditional operations applied before the measurement of . The conditional operation , and therefore , depends only on . Splitting the other term along the partition, this implies

(26)
(27)

which proves that can be obtained from and . As a result, we get

(28)
(29)
(30)

because is independent of and .

Injecting this in Eq. (23), we find