Approximate Relational Reasoning for Quantum Programs

Abstract

Quantum computation is inevitably subject to imperfections in its implementation. These imperfections arise from various sources, including environmental noise at the hardware level and the introduction of approximate implementations by quantum algorithm designers, such as lower-depth computations. Given the significant advantage of relational logic in program reasoning and the importance of assessing the robustness of quantum programs between their ideal specifications and imperfect implementations, we design a proof system to verify the approximate relational properties of quantum programs. We demonstrate the effectiveness of our approach by providing the first formal verification of the renowned low-depth approximation of the quantum Fourier transform. Furthermore, we validate the approximate correctness of the repeat-until-success algorithm. From the technical point of view, we develop approximate quantum coupling as a fundamental tool to study approximate relational reasoning for quantum programs, a novel generalization of the widely used approximate probabilistic coupling in probabilistic programs, answering a previously posed open question for projective predicates.

1 Introduction

Program equivalence [11, 18, 41] is a central concept in many areas of computer science, including software engineering [31, 36, 54], translation validation of compilers [38], program optimization [30], and program analysis [6, 15, 35]. Relational verification aims to prove the relational properties between two programs. A typical Hoare-style relational judgment is of the form $\vdash c_1 \sim c_2: \varPsi \Rightarrow \varPhi$ where $c_1$ and $c_2$ represent two compared programs, $\varPsi$ and $\varPhi$ are relational assertions in the deterministic scenario [10], where relational Hoare logic (RHL) predicates are binary relations over memories. The judgment states that for any initial memories $m_1$ and $m_2$ that satisfy the precondition $\varPsi$, the resulting memories $m_1'$ and $m_2'$ should satisfy postcondition $\varPhi$. For probabilistic programs [7], probabilistic relational Hoare logic (pRHL) lifted the predicates into relations over probabilistic distributions on memories. Furthermore, [9] introduced extra parameters to allow approximate lifting of relations to distributions. To be specific, the judgments defined in approximate probabilistic relational Hoare logic (apRHL) are of the form $c_1 \sim _{\alpha ,\delta } c_2 : \varPsi \Rightarrow \varPhi$ with parameters $\alpha ,\delta$ for reasoning about differential privacy.

Since the emergence of quantum programming languages, there have been various works [25, 27, 32, 44, 57, 59,60,61,62] about the formal verification of quantum programs. Among techniques in program analysis, the exact relational logic for quantum programs attracts lots of attention [8, 33, 51]. Relational logic provides a more expressive approach to characterize the relation between two programs. For instance, direct verification of the equivalence between quantum programs $S_1$ and $S_2$ defined on register $\bar{q}$ requires checking the equivalence between $\llbracket S_1 \rrbracket (\rho )$ and $\llbracket S_2 \rrbracket (\rho )$ for all $\rho$ in Hilbert space $\mathcal {H}_{\bar{q}}$ that involves enumerations of an infinite set. A quantum relational judgment concerning the quantum equivalence predicate can concisely explain the direct enumerations. However, none of the above works considers approximate reasoning that is universal in practice.

• It is implausible to physically implement quantum gates with perfect accuracy on the hardware level, and the need to consider approximations is likely inevitable. As noted by John Preskill, the noise in quantum gates will limit the size of reliable quantum circuits, and technologies for more accurate quantum gates are of great value in the Noisy Intermediate-Scale Quantum (NISQ) [42] era.

• On the software level, the NISQ nature of hardware signifies the importance of taking noise into account at the level of quantum algorithm design. More specifically, approximate computation can be more efficient and less erroneous than precise one since it can improve the depth of circuits and simplify the calculation. A good example is the approximate quantum Fourier transform [16], which achieves a lower circuit depth approximation of the exact quantum Fourier transform used in Shor’s celebrated algorithm [45].

As for approximate reasoning in quantum settings, [66] discussed the robustness of quantum programs by introducing the concept of approximate satisfaction of predicates, [26] proposed a parameterized diamond norm to characterize the distance between an ideal program and a noisy one. Despite the significant advancements in quantum approximate reasoning and the recognition of the importance of relational reasoning, there remains a notable gap in the field — an absence of a robust logical framework for effectively reasoning about the relational properties between quantum programs approximately. In quantum approximate relational reasoning, the main obstacles are:

• There is no mathematical theory for a quantum version of approximate couplings, an open question in [8]. The lack of such a theory significantly affects the applications of exact quantum coupling and relations quantum Hoare logic. Usually, two quantum programs have different branching probabilities in the presence of noise or approximations. Under these circumstances, their corresponding quantum states have different traces, where exact quantum couplings on these states do not exist. The main difficulties in defining an approximate quantum coupling include defining a distance between quantum states, which can be highly nonlinear. Previous knowledge about probabilistic couplings may not directly apply: even for the exact quantum coupling, fundamental properties of probabilistic coupling [24] are no longer true [67].

• Designing an approximate relational quantum Hoare logic system is indeed highly challenging. The system needs to consider several factors, including infinite executions of quantum while loops, approximated quantum couplings, and the applicability of the logic rules. In quantum programming, a while loop can have infinite executions of the loop body because of the probabilistic feature of quantum measurements. Furthermore, when dealing with approximate quantum couplings, the system must handle the inherent uncertainty and approximation errors that arise when coupling with program branches.

• The applicability of logic rules adds another layer of complexity. To strike a balance between the accuracy of the logic rules and simplicity, efficiency, and usability is a crucial consideration when designing a logic system. Ensuring the logic rules are powerful yet easy to use for reasoning relational properties of complicated quantum programs requires careful consideration and analysis.

In this paper, we derive an approximate version of the existing quantum relational Hoare logic, thus making approximate relational reasoning feasible. Our judgment is of the form

$$\begin{aligned} S_1 \sim _{\delta } S_2 : A \Rightarrow B \end{aligned}$$

where $S_1$ and $S_2$ represent compared quantum programs, A and B are projective quantum predicates over the whole system. The validity of our judgment is based on the idea of approximate (quantum) coupling and lifting. A state $\rho$ is a $\delta$-coupling for the state pair $\langle \rho _1, \rho _2 \rangle$ if trace distances $D(\rho _1, \textrm{Tr}_2(\rho ))$ and $D(\rho _2, \textrm{Tr}_1(\rho ))$ are both not bigger than $\delta$. A state $\sigma$ is a witness of the $\delta$-lifting $\rho _1 \sim _P^\delta \rho _2$ if $\sigma$ is a $\delta$-coupling for the state pair $\langle \rho _1, \rho _2 \rangle$ and satisfies the predicate P ($P\sigma = \sigma$). Informally, our judgment holds if for any quantum lifting $\rho _1 \sim _A^0 \rho _2$ of the inputs, there exists a witness of the $\delta$-lifting $\llbracket S_1 \rrbracket (\rho _1) \sim _P^\delta \llbracket S_2 \rrbracket (\rho _2)$ of the outputs.¹

Technical contributions include:

• Approximate quantum liftings. We propose a novel notion of approximate quantum liftings concerning projection-based quantum predicates to make approximate reasoning simple and powerful. We do not require two quantum states to have the same trace in approximate quantum lifting. In other words, the exact quantum coupling may not exist. We employ the existing distances, including trace distance and diamond norm, and define a “Hausdorff-like” distance between projections incorporated with quantum coupling to be the metric of the approximation of the couplings.

• Sound aqRHL. We propose a formal relational judgment to incorporate the spirit of classical apRHL with a new quantum explanation based on the proposed approximate quantum liftings. A sound approximate quantum relational Hoare logic (aqRHL) is built based on our relational judgments. Our choice of quantum $\delta$-lifting allows us to track the relational properties of two programs with different classical branching probabilities. In particular, our methodology allows us to compute proper upper bounds for approximate liftings for quantum equivalence relations, which plays a central role in characterizing the equivalence of quantum programs.

• Application. We demonstrate the first formal verification of the low-depth approximate quantum Fourier transform (QFT) with an error bound that is asymptotically equivalent to the one in [16]. Implementing QFT is a significant step in the development of quantum algorithms such as period finding [45], HHL algorithm [21] and quantum principal component analysis [34]. We also apply aqRHL, particularly the loop rule, to reason the repeat until success which is one of the essential loop programs in quantum computation. Other applications covered in the complete edition of this paper include the verification of appropriate decomposition of unitary gates, and the correctness of bit flip code against an arbitrary single-qubit error.

2 Preliminary and Notations

This section offers a brief introduction to quantum computation and necessary notations from [39].

The state space of a quantum system is a Hilbert space $\mathcal {H}$. The Dirac notation $|\psi \rangle$ denotes a unit complex vector (called pure state or vector state). The most important orthonormal basis of one-qubit system is the computational basis, i.e. $\{|0\rangle , |1\rangle \}$. Superposition is a key feature that makes quantum programs different from classical ones, such as a qubit being in the superposition $(|0\rangle \pm |1\rangle )/\sqrt{2}$. An operator acting on an d-dimensional Hilbert space $\mathcal {H}$ is represented as a $d \times d$ matrix. A positive semi-definite, Hermitian operator, $\rho$ acting on $\mathcal {H}$, is called a partial density operator if its trace satisfies $\textrm{Tr}(\rho ) \le 1$. Particularly, $\rho$ is called a density operator if $\textrm{Tr}(\rho )=1$. The partial density operators can represent both pure and mixed quantum states. For a pure state $|\psi \rangle$, its partial density operator is $|\psi \rangle \langle \psi |$, where $\langle \psi |$ is the conjugate transpose of $|\psi \rangle$. For a mixed state which is a classical distribution $\{p_i\}$ over pure states $\{|\psi _i\rangle \}$, its partial density operator is $\sum _i p_i |\psi _i\rangle \langle \psi _i|$. We use $\mathcal {D}(\mathcal {H})$ to denote the set of all partial density operators acting on Hilbert space $\mathcal {H}$.

Let $\bar{q}_1$ and $\bar{q}_2$ be two independent registers in states $\rho _1 \in \mathcal {D}(\mathcal {H}_{\bar{q}_1})$ and $\rho _2 \in \mathcal {D}(\mathcal {H}_{\bar{q}_2})$ respectively, the composite register $\bar{q} = \{\bar{q}_1, \bar{q}_2\}$ is then in the state $\rho _1 \otimes \rho _2 \in \mathcal {H}_{\bar{q}} = \mathcal {H}_{\bar{q}_1} \otimes \mathcal {H}_{\bar{q}_2}$. Partial trace is a very useful tool for describing subsystems of a composite quantum system. Formally, the partial trace over $\mathcal {H}_{\bar{q}_1}$ is a mapping $\textrm{Tr}_1(\cdot )$ from operators in $\mathcal {H}_{\bar{q}_1} \otimes \mathcal {H}_{\bar{q}_2}$ to operators in $\mathcal {H}_{\bar{q}_2}$ defined by $\textrm{Tr}_1(|\varphi _1\rangle \langle \psi _1| \otimes |\varphi _2\rangle \langle \psi _2|) = \langle \psi _1|\varphi _1\rangle \cdot |\varphi _2\rangle \langle \psi _2|$ for any $|\psi _1\rangle , |\varphi _1\rangle \in \mathcal {H}_{\bar{q}_1}$ and $|\psi _2\rangle , |\varphi _2\rangle \in \mathcal {H}_{\bar{q}_2}$. The partial trace $\textrm{Tr}_2(\cdot )$ can be defined symmetrically. If the composite system $\bar{q} = \{\bar{q}_1, \bar{q}_2 \}$ is in the state $\rho$, then subsystems $\bar{q}_1$ and $\bar{q}_2$ are in states $\textrm{Tr}_2(\rho )$ and $\textrm{Tr}_1(\rho )$ respectively.

The evolution of an isolated quantum system can be characterized by a unitary operator U such that $U^\dagger U = U U^\dagger = I$, where $\dagger$ denotes the conjugate and transpose. Here we introduce some commonly used unitary operators, also known as “gates”, that will be used in later examples:

$$\begin{aligned}\begin{gathered} X = \begin{pmatrix} 0 &{} 1 \\ 1 &{} 0 \end{pmatrix} \quad Y = \begin{pmatrix} 0 &{} -i \\ i &{} 0 \end{pmatrix} \quad Z = \begin{pmatrix} 1 &{} 0 \\ 0 &{} -1 \end{pmatrix} \quad H = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 &{} 1\\ 1 &{} -1 \end{pmatrix} \\ P(\theta ) = \begin{pmatrix} 1 &{} 0 \\ 0 &{} e^{i \theta } \end{pmatrix} \quad \textit{CNOT}= \begin{pmatrix} I &{} 0 \\ 0 &{} X \end{pmatrix} \quad c\text {-}P(\theta ) = \begin{pmatrix} I &{} 0 \\ 0 &{} P(\theta ) \end{pmatrix} \end{gathered}\end{aligned}$$

The act of extracting information from a quantum system is known as quantum measurement. A measurement $\mathcal {M}=\{M_m\}$ is described by a set of linear operators over $\mathcal {H}$ such that $\sum _m M_m^\dagger M_m = I$, where the subscript m refers to the measurement outcome. Applying a quantum measurement $\mathcal {M}$ on $|\psi \rangle$, the probability of observing outcome m is $p_m=\langle \psi |\!M_m M_m^\dagger \!|\psi \rangle$, and the state after the measurement collapses into $M_m|\psi \rangle /\sqrt{p_m}$.

A projection is a linear operator P on $\mathcal {H}$ that satisfies $P^2=P=P^\dagger$. This paper adopts the convention from [12] and recent work [51, 66] that constrains quantum predicates to be Hilbert spaces or projections. The complete partial order over Hilbert subspaces is equivalent to the inclusion relation $\subseteq$. This choice of predicates enables us to define the assertion about quantum states.

Definition 1

(Support). If $A=\sum _i \lambda _i |\psi _i\rangle \langle \psi _i|$, where $|\psi _i\rangle$ is an unit vector in $\mathcal {H}$ and $\lambda _i>0$, then the support of A is the space spanned by $\{|\psi _i\rangle \}$. I.e., $\textrm{supp}(A) = \textrm{span}\{|\psi _i\rangle \}$.

Definition 2

(Satisfaction). A partial density operator $\rho$ satisfies a predicate P, denoted by $\rho \vDash P$, if $\textrm{supp}(\rho ) \subseteq P$.

A general quantum operation, described by a superoperator $\mathcal {E}$, can be implemented by combining unitary transformations with quantum measurements by introducing ancilla systems and discarding post-measurement states. A superoperator always maps density matrices to partial density matrices and has the Kraus representation. Readers may refer to the system-environment model in Sect. 8.2 [39] for more details.

3 Quantum Programming Language

In this section, we review the syntax and semantics of the quantum while-language [59]. We use var(S) to represent the set of all variables present in a quantum program S, and $\mathcal {H}_S = \otimes _{q \in var (S)} \mathcal {H}_q$ to denote the Hilbert space of all the quantum variables in program S. The syntax in Definition 3 is the same as [59] except that the conditional statement is replaced by the $\textbf{if}$ statement.

Definition 3

(Syntax). The following syntax defines the quantum programs:

$$\begin{aligned} (Stmts ) \quad S \ \,\,{:}{:}\!\!= & \ \textbf{skip}\mid q:=|0\rangle \mid \bar{q}:=U[\bar{q}] \mid S_1;S_2 \\ & \mid \textbf{if}\ (\square m \cdot \mathcal {M}[\bar{q}]=m \rightarrow S_m) \ \textbf{fi}\mid \textbf{while}\ \mathcal {M}[\bar{q}]=1 \ \textbf{do}\ S \ \textbf{od}\end{aligned}$$

The denotational semantics of the quantum while-language are presented in Fig. 1. By convention, we use $\llbracket S \rrbracket$ to denote the semantic function of a program S. Statement $q:=|0\rangle$ initializes a variable q in state $\rho$ to $|0\rangle \langle 0|$ while leaving other variables unchanged, where $|\psi \rangle _q\!\langle \varphi |$ denote the outer product of the vector states $|\psi \rangle$ and $|\varphi \rangle$ in $\mathcal {H}_q$. The statement $\bar{q}:= U[\bar{q}]$ performs the unitary transition $\rho \mapsto U \rho U^\dagger$ over register $\bar{q}$. Quantum measurements work as guards to set a variable in a mixed state. For the if statement, if the measurement outcome is m, the input state $\rho$ will collapse into $M_m \rho M_m^\dagger /p_m$ with probability $p_m = \textrm{Tr}(M_m \rho M_m^\dagger )$ and then executes subprogram $S_m$. Here we absorb the probability $p_m$ into the collapse state, and $M_m \rho M_m^\dagger$ represents the corresponding measurement output. The final output is the summation of the outputs of all branches. For the loop statement, $\mathcal {M}_0 \circ (\llbracket S \rrbracket \circ \mathcal {M}_1)^k$ denotes the k-th unrolling of loop statement.

Fig. 1.

Denotational semantics of quantum while-language

Lemma 1

([59]). For any quantum while program S defined in Fig. 1, its denotational semantics function $\llbracket S \rrbracket :\mathcal {D}(\mathcal {H})\mapsto \mathcal {D}(\mathcal {H})$ is a superoperator.

4 Quantum Approximate Coupling and Liftings

4.1 Approximate Quantum Coupling and Lifting

We first review the quantum generalization of the classical trace distance, a commonly used metric for the difference between two (partial) quantum states.

Definition 4

The trace distance of two partial density operators $\rho$ and $\sigma$ is $D(\rho , \sigma ) \ \equiv \ \frac{1}{2}\textrm{Tr}|\rho - \sigma |$, where $|A| = \sqrt{A^\dagger A}$ for any operator A, i.e., the positive square root of $A^\dagger A$.

In the classical setting, two discrete distributions $\mu _1$ and $\mu _2$ over sets $A_1$ and $A_2$ are coupled by a distribution $\mu$ over $A_1 \times A_2$ if and only if the first and second marginals of $\mu$ are exactly $\mu _1$ and $\mu _2$, respectively. This notion of coupling for distributions naturally generalizes to an exact quantum coupling [8] for density matrices. Formally, we say $\rho$ is an exact coupling for $\langle \rho _1, \rho _2 \rangle$ if $\ \textrm{Tr}_1(\rho ) = \rho _2$ and $\textrm{Tr}_2(\rho )=\rho _1$. Commonly, the quantum measurements in two quantum programs produce different probability distributions. In such cases, exact quantum coupling does not exist between branches. We propose approximate quantum coupling parameterized by deviation $\delta$ to bound the trace distance between the “marginal” density matrices.

Definition 5

(Approximate Quantum Coupling). Let $\rho _1 \in \mathcal {D}(\mathcal {H}_1)$ and $\rho _2 \in \mathcal {D}(\mathcal {H}_2)$, then $\rho \in \mathcal {D}(\mathcal {H}_1 \otimes \mathcal {H}_2)$ is a $\delta$-coupling for $\langle \rho _1, \rho _2 \rangle$ if

$$\begin{aligned} D(\rho _1,\textrm{Tr}_2(\rho )) \le \delta \qquad D(\rho _2,\textrm{Tr}_1(\rho )) \le \delta \end{aligned}$$

The approximate quantum coupling degenerates to the exact version if $\delta = 0$. Like the classical case, approximate quantum coupling induces approximate semantics of projective predicates via approximate lifting.

Definition 6

(Approximate Quantum Lifting). Let $\rho _1 \in \mathcal {D}(\mathcal {H}_1)$ and $\rho _2 \in \mathcal {D}(\mathcal {H}_2)$, let P be a projection onto a closed subspace of $\mathcal {H}_1 \otimes \mathcal {H}_2$, then $\rho \in D(\mathcal {H}_1 \otimes \mathcal {H}_2)$ is called a witness of the $\delta$-lifting $\rho _1 \sim _P^\delta \rho _2$ if,

• 1. $\rho$ is a $\delta$-coupling for $\langle \rho _1, \rho _2 \rangle$;

• 2. $\textrm{supp}(\rho ) \subseteq P$.

where $\delta$ is the deviation from the exact quantum lifting.

A valid approximate quantum lifting implies the existence of an approximate quantum coupling that satisfies a quantum predicate. The approximate lifting $\rho _1 \sim _P^\delta \rho _2$ degenerates into the exact lifting $\rho _1 \sim _P \rho _2$ when $\delta = 0$. One of the most important quantum predicates is the equivalence relation between two registers, as defined below [8].

Definition 7

(Equivalence). Let register $\bar{p}$ and $\bar{q}$ are two disjoint registers of the same size. The quantum equivalence predicate over $(\bar{p}, \bar{q})$, denoted by $\equiv _{(\bar{p}, \bar{q})}$, is the projection

$$\begin{aligned} (I_{\bar{p}} \otimes I_{\bar{q}} + \textit{SWAP})/2 \end{aligned}$$

over subspace $\mathcal {H}_{\bar{p}} \otimes \mathcal {H}_{\bar{q}}$. $\textit{SWAP}$ is the swap operator defined on $(\bar{p}, \bar{q})$ such that by $\textit{SWAP}|\psi \rangle |\varphi \rangle = |\varphi \rangle |\psi \rangle$ for any $|\psi \rangle \in \mathcal {H}_{\bar{p}}$ and $|\varphi \rangle \in \mathcal {H}_{\bar{q}}$.

The quantum equivalence predicate in Definition 7 directly comes from a natural observation. In the probabilistic world, if two probability distributions $\mu _1$ and $\mu _1$ over X are the same, then there exists a coupling $\mu$ whose support lives in the identity relation $\{(a, a) \mid a \in X \}$. In quantum settings, this is not true due to superposition. For example, the exact coupling of the state $|+\rangle = (|0\rangle +|1\rangle )/\sqrt{2}$ and itself is $|+\rangle \otimes |+\rangle$, which is not in the space spanned by $|0\rangle \otimes |0\rangle$ and $|1\rangle \otimes |1\rangle$. Instead, we use the projection $(I+ \textit{SWAP})/2$ to represent the corresponding symmetric space. By doing so, we have $(I+ \textit{SWAP})(|+\rangle \otimes |+\rangle )/2 = |+\rangle \otimes |+\rangle$. The following lemma shows that approximate lifting concerning $\equiv$² effectively encodes the trace distance of two partial density matrices.

Lemma 2

For any $\rho _1, \rho _2$, we have $\rho _1 \sim _\equiv ^\delta \rho _2 \ \Leftrightarrow \ D(\rho _1, \rho _2) \le 2\delta$. Particularly, if $\delta = 0$, we have $\rho _1 \sim _\equiv \rho _2 \ \Leftrightarrow \ \rho _1=\rho _2$.

The introduction of approximate couplings/liftings is necessary when the comparison can not match the desired predicate exactly. For example, the implementation of a unitary gate U can be approximated by a proper RUS circuit [13, 40]. Given an input $\rho$, quantum measurement in each iteration of the RUS circuit will generate the desired output $U \rho U^\dagger$ with a probability p, where p is determined by the construction of the RUS circuit. If the iterations of the RUS circuit are unbounded, the desired state can eventually be achieved. The RUS algorithm’s function $\mathcal {E}$ converges to U, as expressed as $\mathcal {E}(\rho ) = \sum _{k=1}^{\infty } p(1-p)^{k-1} U \rho U^\dagger = U \rho U^\dagger$. In this case, the exact lifting can accurately describe the equivalence $\mathcal {E}(\rho ) \sim _\equiv U \rho U^\dagger$ with no problem. However, in practical scenarios, there typically exists an upper bound N on the iteration count k, leading to an approximate equivalence denoted as $\mathcal {E}'(\rho ) \sim _\equiv ^\delta U \rho U^\dagger$, where $\mathcal {E}'$ represents the function with bounded looping and $\delta = (1-p)^N/2$.

4.2 Upper Bound of Approximation

The approximation usually arises when we use a desired postcondition to approximate an exact postcondition. Formally, let (A, B) be the pair of two projections A and B over Hilbert space $\mathcal {H}_1 \otimes \mathcal {H}_2$, the inference

$$\begin{aligned} \forall \rho _1, \rho _2, \rho _1 \sim _A \rho _2 \ \Rightarrow \ \rho _1 \sim _B^{\delta } \rho _2 \end{aligned}$$

(1)

demonstrates a general way of introducing approximate reasoning. That is, given a witness of the exact lifting $\rho _1 \sim _A \rho _2$, does there exist a witness $\sigma$ of the approximate lifting $\rho _1 \sim _B^{\delta \rho _2}$?. The optimal deviation $\delta$ in Eq. 1 is equivalent to the following quantity,

$$\begin{aligned} \delta = d(A, B) = \sup \limits _{\rho \vDash A} \ \inf \limits _{\sigma \vDash B} \max \{ D(\textrm{Tr}_1(\rho ), \textrm{Tr}_1(\sigma )), D(\textrm{Tr}_2(\rho ), \textrm{Tr}_2(\sigma )) \} \end{aligned}$$

(2)

where d(A, B) can be upper bounded by $\sup \limits _{\rho \vDash A} \ \inf \limits _{\sigma \vDash B} D(\rho , \sigma )$ introduced in [66].

In the following, we discuss a simple but important instance of Eq. 1 with $A = (U_1 \otimes U_2) B (U_1 \otimes U_2)^\dagger$ and B being the quantum equivalence predicate $\equiv$. Then Eq. 1 can be represented as follows,

$$\begin{aligned} \forall \rho _1, \rho _2, \rho _1 \sim _\equiv \rho _2 \ \Rightarrow \ U_1 \rho _1 U_1^\dagger \sim _{A} U_2 \rho _2 U_2^\dagger \ \Rightarrow \ U_1 \rho _1 U_1^\dagger \sim _{\equiv }^{\delta } U_2 \rho _2 U_2^\dagger \end{aligned}$$

where $\delta$ can be upper bounded by $\Vert U_1 \cdot U_1^\dagger - U_2 \cdot U_2^\dagger \Vert _\diamond$. The diamond norm $\Vert \cdot \Vert _\diamond$ proposed by Kitaev [2] can better distinguish between two superoperators with the help of the power of quantum entanglement by introducing auxiliary qubits.

Definition 8

(Diamond Norm). Let $\mathcal {A}:\mathcal {L}(\mathcal {H})\mapsto \mathcal {L}(\mathcal {H})$ with $\mathcal {L}(\mathcal {H})$ denoting the matrix space of $\mathcal {H}$,

$$\begin{aligned} \Vert \mathcal {A} \Vert _\diamond \ \equiv \ \max _{\rho \in \mathcal {D}(\mathcal {H} \otimes \mathcal {H}')} \frac{1}{2}\textrm{Tr}|(\mathcal {A} \otimes I_{\mathcal {H}'})(\rho )| \end{aligned}$$

(3)

where $\ \mathcal {H}'$ denotes a copy of $\ \mathcal {H}$. The factor 1/2 is added to keep consistent with trace distance.

It is straightforward to verify that $D(\mathcal {E}_1(\sigma ),\mathcal {E}_2(\sigma )) \le \Vert \mathcal {E}_1 - \mathcal {E}_2 \Vert _\diamond$ for any $\sigma \in \mathcal {D}(\mathcal {H})$ for superoperators $\mathcal {E}_1, \mathcal {E}_2$ by choosing $\rho = \sigma \otimes I_{\mathcal {H}'}/2^{Dim(\mathcal {H}')}$.³ The distance between two superoperators can be computed efficiently [53]. Particularly, the distance between $U_1$ and $U_2$ is

$$\begin{aligned} \textstyle \Vert U_1 \cdot U_1^\dagger - U_2 \cdot U_2^\dagger \Vert _\diamond = {\left\{ \begin{array}{ll} \sin {\alpha /2} \quad \qquad &{}\alpha < \pi \\ 1 &{}\alpha \ge \pi \end{array}\right. } \end{aligned}$$

(4)

where $\alpha$ is the smallest arc containing the spectrum of $U_1^\dagger U_2$ [37].

5 Approximate Relational Logic

5.1 Judgment and Validity

Our logic, called aqRHL, “approximates” the quantum relational Hoare logic described in [8]. The judgments in aqRHL take the following form $S_1 \sim _{\delta } S_2 : A \Rightarrow B$, where $S_1$ and $S_2$ are quantum programs, A and B are projections over subspaces of $\mathcal {H}_{\bar{q}_1}\otimes \mathcal {H}_{\bar{q}_2}$ such that $\bar{q}_i$ contains all free variables of $S_i$, $\delta \in [0,1/2]$ is referred to as the deviation from the exact quantum lifting, respectively. Registers $\bar{q}_1$ and $\bar{q}_2$ are often omitted since they rarely change along our reasoning and are often clear from the context.

Definition 9

(Validity). The approximate relational judgement $S_1 \sim _{\delta } S_2 : A \Rightarrow B$ is valid, written as $\ \vDash S_1 \sim _{\delta } S_2 : A \Rightarrow B$, if and only if

where A and B are projections. If the deviation $\delta$ equals zero, it will be omitted for simplicity.

In Definition 9, we choose projective predicates [12] over the joint system of two programs because such predicates are the quantum analog of binary relations, the predicates used in pRHL [9]. Moreover, this definition will become a judgment of [8] if $\delta = 0$. One of the most important applications of relational Hoare logic is to verify the equivalence between programs, as presented in the following lemma. Naturally, the approximate equivalence between programs can also be reasoned by the approximate relational judgment, where $\delta$ characterizes the deviation of approximation.

Lemma 3

(Program Equivalence). Program $S_1$ is equivalent⁴ to program $S_2$ if and only if $\ \vDash S_1 \sim S_2: \ \equiv \ \Rightarrow \ \equiv$.

The program equivalence can be expressed concisely with predicates being the equivalence relation, instead of checking whether two quantum programs perform uniformly by enumeration of an infinite number of states in Hilbert space. The following example shows that superposition makes quantum program equivalence more complex than its classical counterpart.

Example 1

Let $S_1$ and $S_2$ be two programs defined on a single bit or qubit. For classical programs, the state space for programs $S_1$ and $S_2$ is the set $\{|0\rangle , |1\rangle \}$. Let $\varPsi$ and $\varPhi$ be the equivalence relation, the relational judgment $S_1 \sim S_2: \varPsi \ \Rightarrow \ \varPhi$ holds for classical programs $S_1$ and $S_2$ if

$$\begin{aligned} \llbracket S_1 \rrbracket (|0\rangle \langle 0|) = \llbracket S_2 \rrbracket (|0\rangle \langle 0|) \qquad \llbracket S_1 \rrbracket (|1\rangle \langle 1|) = \llbracket S_2 \rrbracket (|1\rangle \langle 1|) \end{aligned}$$

(5)

However, this conclusion no longer holds in quantum programs since the input state could be a superposition of $|0\rangle$ and $|1\rangle$. For example, let $S_1 \,\,{:}{:}\!\!= \textbf{skip}$ and $S_2 \,\,{:}{:}\!\!= q:= Z[q]$, it is clear that $S_1$ and $S_2$ are not equivalent⁵ although Eq. 5 still holds. To check quantum program equivalence, we need to verify the validity of $\ \llbracket S_1 \rrbracket (\rho ) = \llbracket S_2 \rrbracket (\rho )$ for all $\rho$ in the Hilbert space $\textrm{span}\{|0\rangle , |1\rangle \}$ rather than the set $\{|0\rangle , |1\rangle \}$, which involves enumerations of an infinite set.

5.2 Proof Rules

We are ready to provide some proof rules for our $\text {aqRHL}$ judgments. These rules include construct-specific rules (two-sided and one-sided) and structural ones, as is typical in relational Hoare logic. Notice that rules for branching structures are discussed in Sect. 7 later.

Simple Rules. Figure 2 includes the two/one-sided proof rules for basic statements and sequence structure. The basic rules, namely [Skip], [Init], [Ut] are similar to their counterparts in [8] with $\delta = 0$, where they are presented in the forward variant. Here we use $\textrm{proj}(A)$ to lift non-projection A to its support before assigning it as a predicate. Notice that the rule [Ut] gives the strongest postcondition, which means the reverse $\vdash \bar{q}_1:=U_1^{-1}[\bar{q}_1] \sim \bar{q}_2:=U_2^{-1}[\bar{q}_2]: (U_1 \otimes U_2)A(U_1^\dagger \otimes U_2^\dagger ) \Rightarrow A$ still holds. The rule [Seq] demonstrates that the deviation grows linearly with respect to the sequences, which directly comes from the triangle inequality of trace distance. One-sided rules are necessary when two programs do not share the same structure. We have only listed the one-side rules (appended with “-L”) for the left side, and similar rules apply to the right side symmetrically.

Fig. 2.

Simple $\text {aqRHL}$ rules.

Rules for Equivalence Relation.

We address a scenario regarding the rules [Ut], where the precondition and postcondition are equivalence relations defined in Definition 7. We use diamond norm to bound the deviation in rule [Ut-id], where $U \cdot U^\dagger$ denotes the Kraus representation [39] of unitary U. The rule [Comp] permits reasoning equivalence between programs by introducing intermediate programs (Fig. 3).

Fig. 3.

Rules for Equivalence Relation

5.3 Soundness Theorem

Theorem 1

[Soundness] For any program $S_1,S_2$, projections A and B, deviation $\delta$, we have,

$$\vdash S_1 \sim _{\delta } S_2: A \Rightarrow B \quad \Rightarrow \quad \ \vDash S_1 \sim _{\delta } S_2: A \Rightarrow B$$

The soundness of our proof system is proved with respect to the validity of judgments defined in 9, while the completeness remains an open question. For classical deterministic programs, relational Hoare logic has been demonstrated to be relatively complete for terminating programs with the help of providing additional supplementary one-sided rules. However, relative completeness does not extend to probabilistic programs. As highlighted in [5], the probabilistic coupling method lacks the robustness of the conductance method in demonstrating the rapid mixing of Markov chains. Building upon the work laid by [8], the quantum extension of probabilistic couplings and the introduction of approximation in our judgments further complicate this problem.

6 Approximate Quantum Fourier Transform

Objective. As a quantum analog of the classical discrete Fourier transform, quantum Fourier transform (QFT) [17] performs a linear transformation on quantum states and extracts the periodicity of the amplitudes of quantum states. Due to the imperfectness of quantum gates, the approximate quantum Fourier transform (AQFT) is proposed to improve the circuit depth of QFT for efficiency. Reference [17] proposes a direct AQFT based on ignoring gates related to high-order terms. Cleve and Watrous [16] parallelized the phase estimation procedure to perform AQFT with lower circuit depth. Let $S_\text {QFT}$ and $S_\text {AQFT}$ be the corresponding quantum programs for QFT and AQFT, respectively. This section uses our logic to derive the judgment of form $S_\text {QFT} \sim _{\delta } S_\text {AQFT}: \ \equiv \ \Rightarrow \ \equiv$ to reason about how well AQFT approximates QFT.

Specification. For an n qubit system, QFT on a computational basis state $|x\rangle = |x_1 x_2 \dots x_{n}\rangle$ is defined as the linear operation U such that

$$\begin{aligned} \textstyle U |x\rangle = |\psi _x\rangle = \frac{1}{\sqrt{2^n}} \sum _{y=0}^{N-1} (e^{2\pi i/N})^{x \cdot y} |y\rangle \end{aligned}$$

(6)

where $N=2^n$, $|\psi _x\rangle$ is called a Fourier basis state with respect to state $|x\rangle$, $x \cdot y$ denotes the multiplication between the binary representation of x and y. $|\psi _x\rangle$ can be described as $|\psi _x\rangle = |\mu _{0.x_n}\rangle |\mu _{0.x_{n-1} x_{n}}\rangle \cdots |\mu _{0.x_1 \dots x_n}\rangle$, where $|\mu _\theta \rangle = (|0\rangle + e^{2\pi i \theta }|1\rangle )/\sqrt{2}$, $0.x_i \dots x_j$ denotes the binary fraction $x_i/2 + x_{i+1}/4 + \cdots + x_j/2^{j-i+1}$. State $|\mu _\theta \rangle$ can be obtained by applying the phase shift gate $P(2\pi \theta )$ (mentioned in Sect. 2) on state $|+\rangle = (|0\rangle +|1\rangle )/\sqrt{2}$. The phase shift gate $P(2\pi \theta )$ can be decomposed as the sequence of gates $R_m = P(2\pi /2^m)$ since $P(\theta _1)P(\theta _2)= P(\theta _1+\theta _2)$. The controlled $R_m$ gate is denoted by $CR_m[(q_1,q_2)]$, which is the $c\text {-}P(\theta )$ gate (mentioned in Sect. 2) with $\theta = 2\pi /2^m$.

QFT can be parallelly implemented [16], as shown in Fig. 4. The unitary V generates the Fourier basis state $|\psi _x\rangle$ without erasing $|x\rangle$. The unitary Add introduces auxiliary $(k-1)n$ qubits to create $k-1$ replicas of Fourier basis state $|\psi _x\rangle$. The unitary oracle T introduces auxiliary n qubits to compute the corresponding phase parameter $|x\rangle$ of the Fourier basis state $|\psi _x\rangle$ without erasing $|\psi _x\rangle$. All these auxiliary qubits are not depicted in Fig. 4 since they are reset back to $|0\rangle$ after the computation.

Fig. 4.

QFT circuit in [16]. Given a computational basis state $|x\rangle$ and corresponding Fourier basis state $|\psi _x\rangle$, unitary V performs mapping $|x\rangle |0\rangle ^{\otimes n} \mapsto |x\rangle |\psi _x\rangle$, unitary Add performs mapping $|\psi _x\rangle |0\rangle ^{\otimes n} \cdots |0\rangle ^{\otimes n} \mapsto |\psi _x\rangle |\psi _x\rangle \cdots |\psi _x\rangle$, and unitary T performs mapping $|\psi _x\rangle \cdots |\psi _x\rangle |0\rangle ^{\otimes n} \mapsto |\psi _x\rangle \cdots |\psi _x\rangle |x\rangle$.

We can perform approximate computations for oracles V and T to achieve a lower circuit depth. Oracle V can be approximated by ignoring $CR_m$ gates of larger m. Oracle T can be approximated by performing quantum measurements followed by classical post-processing on measurement outcomes [28]. Let unitary $V'$ and $T'$ be the approximation of V and T respectively, the corresponding program $S_\text {AQFT}$ is almost the same as $S_\text {QFT}$ but with oracles V and T replaced by $V'$ and $T'$ respectively. Next, we use our logic to reason the approximate equivalence between programs $S_\text {QFT}$ and $S_\text {AQFT}$. That is,

$$\begin{aligned} S_\text {QFT} \sim _{\delta _1 + 2\delta _2} S_\text {AQFT}: \ \equiv _{(\bar{q}_0,\bar{q}_0')} \otimes |0\rangle \langle 0|_{aux} \Rightarrow \ \equiv _{(\bar{q}_0,\bar{q}_0')} \otimes |0\rangle \langle 0|_{aux} \end{aligned}$$

(7)

where $\delta = n\pi 2^{-k-1} + 2ne^{-k/8}$, $|0\rangle \langle 0|_{aux}$ denotes the tensor product of constant projections $|0\rangle \langle 0|$ over all qubits in other registers except $\bar{q}_0$ and $\bar{q}_0'$. The main proof sketch is shown in Fig. 5.

Fig. 5.

Proof sketch for programs $S_\text {QFT}$ and $S_\text {AQFT}$. To easily refer to predicates, we label each assertion a name on its right.

Create the Fourier Basis State. The computation of unitary U in Eq. 6 can be parallelized by individually preparing every $|\mu _\theta \rangle$ by the following unitary

$$\begin{aligned} Q_{t,i}: |0\rangle ^{\otimes t}|x_{1} \dots x_{n}\rangle \mapsto |\mu _{0.x_{i} \dots x_{i+t-1}}\rangle |0\rangle ^{\otimes t-1}|x_{1} \dots x_{n}\rangle \end{aligned}$$

in [16], where $i+t-1 \le n$, qubits $x_1 \dots x_{i-1}$ and $x_{i+t} \dots x_{n}$ in $|x\rangle$ are not used. The unitary $Q_{t,i}$ acting on register $(\bar{q}, \bar{p})$ can be denoted as,

$$\begin{aligned} U_{GHZ}[\bar{q}];CR_1[(\bar{p}[i],\bar{q}[1])];\dots ;CR_t[(\bar{p}[i+t-1],\bar{q}[t])];U_{GHZ}^\dagger [\bar{q}];H[\bar{q}[1]] \end{aligned}$$

where $U_{GHZ}$ denotes the unitary that generates a GHZ state, that is, $U_{GHZ}|0\rangle ^{\otimes t}$ $=(|0\rangle ^{\otimes t}+|1\rangle ^{\otimes t})/\sqrt{2}$. Registers $\bar{q}$ and $\bar{p}$ are of size t and n, respectively. $\bar{q}[i]$ denotes the i-th qubit in register $\bar{q}$. For example, Fig. 6 in [16] represents the circuit of unitary $Q_{4,i}$ on $|x\rangle$. Similar to the approximation in [17], unitary $Q_{t, i}$ can be approximated by ignoring $CR_m$ gates of large m. That is, we could use $Q_{t,i}$ to approximate $Q_{t',i}$ if $1 \le t < t' \le n$. The approximation can be modeled by the judgment

$$\begin{aligned} \begin{gathered} \vdash (\bar{q}, \bar{p}) := Q_{t,i}[(\bar{q}, \bar{p})] \sim _{\delta (t,t')} (\bar{q}', \bar{p}') := Q_{t',i}[(\bar{q}', \bar{p}')]: \\ |0\rangle \langle 0|_{(\bar{q},\bar{q}')} \otimes \equiv _{(\bar{p},\bar{p}')} \ \Rightarrow \ \equiv _{(\bar{q}[1], \bar{q}'[1])} \otimes |0\rangle \langle 0|_{(\bar{q}[2,n],\bar{q}'[2,n])} \otimes \equiv _{(\bar{p},\bar{p}')} \end{gathered} \end{aligned}$$

(8)

with $\delta (t,t') = \frac{1}{2}\sin {\pi (2^{-t}-{2^{-t'}})}$. $P_{\bar{q}}$ denotes a projection P over the register $\bar{q}$. Particularly, $|\psi \rangle \langle \psi |_{\bar{q}}$ denotes the tensor product of $|\psi \rangle \langle \psi |$ over all qubits in register $\bar{q}$.

Figure 7 illustrates the circuit of the oracle V over register $(\bar{q}_0, \bar{r}, \bar{q}_1)$. To prepare each $|\mu _\theta \rangle$ in $|\psi _x\rangle$ individually, we need to prepare n copies of state $|x\rangle$ beforehand, which is achieved by the unitary C. The unitary C can be implemented by CNOT gates in a binary tree architecture to achieve a circuit depth of $\log {n}$. To make it concise, the auxiliary qubits q[2, n] in oracle $Q_{t,i}(\bar{q},\bar{p})$ that reset back to $|0\rangle$ are ignored in Fig. 7 and the input of $Q_{t,i}$ is set as $|0\rangle |x\rangle$. The circuit for oracle $V'$ is almost the same as Fig. 7 except that $Q_{t, i}$ is approximated by $Q_{k, i}$, where k $(0 < k < t \le n)$ denotes the number of significant phase shift gates. Specifically, oracles V and $V'$ can be represented as follows,

$$\begin{aligned} V = \ {} & C[(\bar{q}_0, \bar{r})]; Q_{1,n}[(\bar{q}_1[1],\bar{q}_0)];Q_{2,n-1}[(\bar{q}_1[2],\bar{r}_1)];\dots ;Q_{n,1}[(\bar{q}_1[n],\bar{r}_{n-1})];C^\dagger [(\bar{q}_0, \bar{r})] \\ V' = \ {} & C[(\bar{q}_0', \bar{r}')]; Q_{1,n}[(\bar{q}_1'[1],\bar{q}_0')];Q_{2,n-1}[(\bar{q}_1'[2],\bar{r}'_1)];\dots ;Q_{k,n-k+1}[(\bar{q}_1'[k],\bar{r}'_{k-1})];\\ & Q_{k,n-k+1}[(\bar{q}_1'[k+1],\bar{r}'_k)];\dots ;Q_{k,n-k+1}[(\bar{q}_1'[n],\bar{r}'_{n-1})];C^\dagger [(\bar{q}_0', \bar{r}')] \end{aligned}$$

where register $\bar{r} = \{\bar{r}_1, \dots , \bar{r}_{n-1}\}$ contains $n-1$ registers $\bar{r}_i$ initialized with $|0\rangle ^{\otimes ^n}$. Based on judgement 8, we have the following judgment

$$\begin{aligned} \begin{aligned} \vdash (\bar{q}_0,\bar{q}_1) := V[(\bar{q}_0,\bar{q}_1)] \sim _{\delta _1} (\bar{q}_0',\bar{q}_1') := V'[(\bar{q}_0',\bar{q}_1')] : P_0 \ \Rightarrow \ P_1 \end{aligned} \end{aligned}$$

(9)

where $\delta _1 = \sum _{i=k+1}^{n} \delta (k,i) = \frac{1}{2} \sum _{i=k+1}^{n} \sin {\pi (2^{-k}-{2^{-i}})} \le n\pi 2^{-k-1}$. Notice that every register $\bar{r}_i$ in Fig. 7 is reset back to $|0\rangle$, thus the predicate $|0\rangle \langle 0|_{(\bar{r},\bar{r}')}$ on register $(\bar{r},\bar{r}')$ can be ignored.

Fig. 6.

Circuit for oracle $Q_{4,i}$ on state $|x_1 \dots x_{n}\rangle$. Qubits $x_1 \dots x_{i-1}$ and $x_{i+4} \dots x_{n}$ are not used and ignored.

Fig. 7.

Circuit for oracle V. Given a computational basis state $|x\rangle = |x_1 \dots x_n\rangle$, unitary C performs the mapping $|x\rangle |0\rangle ^{\otimes ^n}\cdots |0\rangle ^{\otimes ^n} \mapsto |x\rangle ^{\otimes ^{n}}$, and unitary $Q_{t,i}$ performs the mapping $|0\rangle |x\rangle \mapsto |\mu _{0.x_{i} \dots x_{i+t-1}}\rangle |x\rangle$.

Replicate & Erase Fourier Basis State. We provide a brief overview of the functionality of the oracle Add as described in [16]. We begin with the state $|\psi _x\rangle |0\rangle ^{\otimes ^n}\cdots |0\rangle ^{\otimes ^n}$ and apply Hadamard gates $H^{\otimes ^n}$ to each $|0\rangle ^{\otimes ^n}$, resulting in $|\psi _x\rangle |\psi _0\rangle \cdots |\psi _0\rangle$. Then, we apply telescoping subtraction $|x_1\rangle |x_2\rangle \cdots |x_k\rangle \rightarrow |x_1\rangle |x_2-x_1\rangle \cdots |x_k - x_{k-1}\rangle$ to obtain $|\psi _x\rangle |\psi _x\rangle \dots |\psi _x\rangle$. Reversely, we can use prefix addition $|x_1\rangle |x_2\rangle \cdots |x_k\rangle \rightarrow |x_1\rangle |x_1+ x_2\rangle \cdots |x_1 + x_2 + \dots + x_{k}\rangle$ to eliminate the duplicates of the Fourier basis state. A log(k)-depth tree of 3-2 adders can be used to generate two encoded numbers, followed by a quantum carry-lookahead adder of log(n)-depth to add the encoded numbers. Since programs $S_\text {AFT}$ and $S_\text {QAFT}$ share the same procedure to replicate and erase Fourier basis states, we simplify replicating and erasing procedures by treating them as quantum oracles Add and $Add^\dagger$ respectively. Then we use rule [Ut] to get the following judgment.

$$\begin{aligned} \vdash (\bar{q}_1,\bar{q}_2) := Add[(\bar{q}_1,\bar{q}_2)] \sim (\bar{q}_1',\bar{q}_2') := Add[(\bar{q}_1',\bar{q}_2')] : P_1 \ \Rightarrow \ P_2 \end{aligned}$$

(10)

$$\begin{aligned} \vdash (\bar{q}_1,\bar{q}_2) := Add^\dagger [(\bar{q}_1,\bar{q}_2)] \sim (\bar{q}_1',\bar{q}_2') := Add^\dagger [(\bar{q}_1',\bar{q}_2')] : P_5 \Rightarrow \ P_6 \end{aligned}$$

(11)

Estimate the Phase of a Fourier State. The key to this step is based on the idea [3] that quantum measurement can be simulated by unitaries with the help of ancillary qubits. As shown in Fig 4, the oracle T generates the phase $|x\rangle$ in register $\bar{q}_3$ of the Fourier state $|\psi _x\rangle$, then the Fourier basis state $|x\rangle$ in register $\bar{q_0}$ can be erased by the following CNOT gate ($CNOT |x\rangle |x\rangle =|x\rangle |0\rangle$). The gate $T^\dagger$, the reverse of T, is applied subsequently to restore the state to the duplicates of $|\psi _x\rangle$. Given the input $|x\rangle$ in register $\bar{q}_0$, the whole process of erasing $|x\rangle$ works as $|x\rangle |\psi _x\rangle \cdots |\psi _x\rangle |0\rangle ^{\otimes ^n} \xrightarrow {T} |x\rangle |\psi _x\rangle \cdots |\psi _x\rangle |x\rangle \xrightarrow {CNOT} |0\rangle ^{\otimes ^n}|\psi _x\rangle \cdots |\psi _x\rangle |x\rangle \xrightarrow {T^\dagger } |0\rangle ^{\otimes ^n}|\psi _x\rangle \cdots |\psi _x\rangle |0\rangle ^{\otimes ^n}$

where the auxiliary register $\bar{q}_3$ is initialized with $|0\rangle ^{\otimes ^n}$ and reset back to $|0\rangle ^{\otimes ^n}$.

In order to reduce the circuit depth of oracle T, [16] parallelized the phase estimation procedure proposed by [29]. Given k copies of each $|\mu _{x2^{-i}}\rangle$, we perform two single-qubit measurements

$$\begin{aligned} \mathcal {M}_1 = \{ M_1^0 = |\mu _{0}\rangle \langle \mu _{0}|, M_1^1 = |\mu _{\frac{1}{2}}\rangle \langle \mu _{\frac{1}{2}}| \} \quad \mathcal {M}_2 = \{ M_2^0 = |\mu _{\frac{1}{4}}\rangle \langle \mu _{\frac{1}{4}}|, M_2^1 = |\mu _{\frac{3}{4}}\rangle \langle \mu _{\frac{3}{4}}| \} \end{aligned}$$

on k/2 of the copies independently, where $\{|\mu _{0}\rangle , |\mu _{\frac{1}{2}}\rangle \}$ and $\{|\mu _{\frac{1}{4}}\rangle , |\mu _{\frac{3}{4}}\rangle \}$ are the eigenvectors of Pauli operators X and Y respectively. These measurements on copies of $|\psi _x\rangle$ would generate a distribution $\{p_{(x,i)}\}$ over a nk-bit string $|m_{(x,i)}\rangle$ of measurement outcomes. Then a reversible classical processing f is applied to infer $x_i'$ based on measurement outcome $|m_{(x,i)}\rangle$, that is $|m_{(x,i)}\rangle |0\rangle \rightarrow |m_{(x,i)}\rangle |x'_i\rangle$, where the probability $p_{(x,i)}$ is close to 1 if $|x'_i\rangle = |x\rangle$, and a properly estimated $|x'_i\rangle$ can be used to erase the phase $|x\rangle$ on register $\bar{q}_0$. The following lemma is proved using Chernoff bound.

Lemma 4

[16] Given any computational basis $|x\rangle$, measuring observables X and Y randomly generates a distribution $\{p_{(x,i)}\}$ over $\{|m_{(x,i)}\rangle \}$, followed by a classical processing that generates phase $|x'_i\rangle$ from $|m_{(x,i)}\rangle$. We have $Pr(|x'_i\rangle = |x\rangle ) = p_{(x,i)} > 1 - 4ne^{-k/8}$.

We can convert the above whole process into a unitary operation $T'$ without actual measurements that can operate on data in superposition. First, the following unitary $U_M([\bar{q}_1,\bar{q}_2,\bar{r}])$,

$$\begin{aligned} U_M([\bar{q}_1,\bar{q}_2,\bar{r}]) := \otimes _{i=1}^n (\otimes _{j=1}^{k/2} U_X[(r[ik+j], p[ik+j])]) \otimes (\otimes _{j=1+k/2}^{k} U_Y[(r[ik+j], p[ik+j])]) \end{aligned}$$

is applied to simulate measurements on copies of $|\psi _x\rangle$, where register $\bar{p} = \{\bar{q}_1,\bar{q}_2\}$ and auxiliary register $\bar{r}$ is initialized with $|0\rangle$. Unitary gates $U_X$ and $U_Y$

$$\begin{aligned} U_X[(q_1,q_2)] &:= (H[q_1] \otimes I[q_2]) CNOT[(q_1,q_2)] (H[q_1] \otimes I[q_2]); \\ U_Y[(q_1,q_2)] &:= (H[q_1] \otimes I[q_2]) CY[(q_1,q_2)] (H[q_1] \otimes I[q_2]) \end{aligned}$$

introduce auxiliary qubit $q_1$ initialized with $|0\rangle$ to simulate single-qubit measurements $\mathcal {M}_1$ and $\mathcal {M}_2$ on $|\mu _{x2^{-i}}\rangle$ in qubit $q_2$.

$$\begin{aligned} |0\rangle |\mu _{x2^{-i}}\rangle &\xrightarrow {U_X} \langle \mu _0|\mu _{x2^{-i}}\rangle \cdot |0\rangle |\mu _0\rangle + \langle \mu _{\frac{1}{2}}|\mu _{x2^{-i}}\rangle \cdot |1\rangle |\mu _{\frac{1}{2}}\rangle \\ |0\rangle |\mu _{x2^{-i}}\rangle &\xrightarrow {U_Y} \langle \mu _{\frac{1}{4}}|\mu _{x2^{-i}}\rangle \cdot |0\rangle |\mu _{\frac{1}{4}}\rangle + \langle \mu _{\frac{3}{4}}|\mu _{x2^{-i}}\rangle \cdot |1\rangle |\mu _{\frac{3}{4}}\rangle \end{aligned}$$

where $CY[(q_1,q_2)]$ denotes the controlled Pauli Y gate. Next, we set the outputs of auxiliary register $\bar{r}$ of $U_M$ to be the input of oracle $O[(\bar{r},\bar{q_3})]$ such that

$$\begin{aligned} U_M |\mu _{x2^{-i}}\rangle \otimes |0\rangle \xrightarrow {O} \textstyle \sum _i \sqrt{p_{(x,i)}} |\varphi \rangle \otimes |x_i'\rangle \end{aligned}$$

where oracle O denotes the corresponding quantum circuit of the classical processing f on measurement outcomes. Thus, the oracle $T'$ can achieved by $U_M[(\bar{q_1},$ $\bar{q_2},\bar{r})]$ and $O[(\bar{r},\bar{q_3})]$ sequentially. By lemma 4, we would have

$$\begin{aligned} \begin{gathered} \vdash (\bar{q}_1,\bar{q}_2,\bar{q}_3) := T [(\bar{q}_1,\bar{q}_2,\bar{q}_3)] \sim _{\delta _2} (\bar{q}_1',\bar{q}_2',\bar{q}_3',\bar{r}) := T' [(\bar{q}_1',\bar{q}_2',\bar{q}_3',\bar{r})] : P_2 \ \Rightarrow \ P_3 \\ \vdash (\bar{q}_1,\bar{q}_2,\bar{q}_3) := T^\dagger [(\bar{q}_1,\bar{q}_2,\bar{q}_3)] \sim _{\delta _2} (\bar{q}_1',\bar{q}_2',\bar{q}_3',\bar{r}) := {T'}^\dagger [(\bar{q}_1',\bar{q}_2',\bar{q}_3',\bar{r})]: P_4 \Rightarrow P_5 \end{gathered} \end{aligned}$$

(12)

where $\delta _2 = 2ne^{-k/8}$.

Conclusion Finally, we use rule [Seq] to sum up all judgments to get Eq. 7.

7 Measurements Conditions and Additional Proof Rules

7.1 Measurement Conditions

Additional constraints must be imposed on programs to establish feasible relational proof rules for those with complex structures, such as if and loop statements. In the classical pRHL approach in [7], the precondition $m_1 \varPsi m_2$ satisfied by the initial memories $m_1$ and $m_2$ requires the guards $e_1$ and $e_2$ in the if or loop statements must be equal. Things get more complex in quantum programs since quantum mechanics are naturally probabilistic, and it is generally impossible to require two if statements to give the same measurement or with the same probability distributions. In [8], the term “synchronous execution” in quantum programs means that two quantum measurements $\mathcal {M}_1 = \{M_1^m\}$ and $\mathcal {M}_2 = \{M_2^m\}$ should produce the same distribution over branches for input $\rho _1$ and $\rho _2$, that is, $\textrm{Tr}(M_1^m \rho _1 {M_1^m}^\dagger ) = \textrm{Tr}(M_2^m \rho _2 {M_2^m}^\dagger )$. To study more general programs, we propose the approximate measurement conditions, which establish appropriate upper bounds for the deviations in our judgments.

Definition 10

(Approximate Measurement Condition). Let $\mathcal {M}_1 = \{M_1^m\}$ and $\mathcal {M}_2 = \{M_2^m\}$ be two measurements in $\mathcal {H}_{1}$ and $\mathcal {H}_{2}$ that share the same set $\{m\}$ of measurement outcomes, respectively. The measurement condition

$$\begin{aligned} \mathcal {M}_1 \approx _{\{\delta _m\}} \mathcal {M}_2 : A \Rightarrow \{(p_m, B_m)\} \end{aligned}$$

(13)

means that for every measurement outcome m, we have

where $p_m \in [0,1]$. Deviations $\delta _m$ in the measurement condition can be ignored if they equal zero. We write predicate $\{(p_m, B_m)\}$ as $\{B_m\}$ for short if all $p_m = 1$.

7.2 Additional Proof Rules

Now, we introduce the rules for if and loop statements in Fig. 8. The rule [IF] requires the measurement condition in the premises to provide a bound on the whole approximation, where the deviation $\delta _m'$ of the branch body is scaled down by $p_m$. The rule [LP] does not require the synchronous execution of loop guards [8, 9], or the speed bound at which loops converge [26]. Instead, the measurement condition only employs an upper bound $p_1$ on the probabilities of entering loop bodies for the first iteration. Rule [LP] requires $p_1 \in [0,1)$ and provides better deviation if $p_1$ is smaller. If $p_1$ equals one initially, we can unroll loop statements several times to make $p_1$ less than one.

We derive rule [LP*] as an alternative to rule [LP] by incorporating more specific measurement conditions for the iterations of loops when we can not find a good A for rule [LP]. When doing approximate reasoning about loops, it is typical to set an upper bound N on the number of iterations. Notice that the factor $\lambda _n$ is not an upper bound on the probability of entering $(n+1)$-th iteration except for $n=0$. Overall, rule [LP*] is a direct application of rule [Seq] on a finite number of iterations, where measurement conditions are used to scale the deviations. Since we can always make the $\textbf{skip}$ statement share the same probability distribution with any $\textbf{if}$ and $\textbf{while}$ statements, the measurement conditions for one-side rules [IF-L], [LP-L], and [LP*-L] are more straightforward.

Fig. 8.

Rules for branching structure in $\text {aqRHL}$. The deviation of rule (LP*) is given by $\textstyle f(\alpha _k,\beta _k,p_k) = (\alpha _0+\sum _{n=0}^{N-1}\lambda _n\alpha _{n+1})+ (N\beta _0+\sum _{n=0}^{N-2}(N-n-1)\lambda _n\beta _{n+1})+(\sum _{n=0}^{N-1}(N-n)\lambda _n\delta _n)$ with $\lambda _n = \prod _{k=0}^n p_k$.

Structural Rules.

Unlike classical programs, the potential quantum entanglement between subsystems brings a unique challenge in constructing a general frame rule for quantum programs [8, 51, 64]. We derive a simple frame rule [Frame] to specify a specific instance that the predicate C on additional independent system $(\bar{r}_1,\bar{r}_2)$ is one-dimensional. Subscripts related to registers are displayed explicitly for clarity. Rule [Order] adds an order relation $\le$ over deviations. In addition, an additional condition is introduced in rule [Approx] to allow switching postconditions at the cost of bringing approximation (Fig. 9).

Fig. 9.

Structural aqRHL rules.

8 Related and Future Works

With the fast development of quantum hardware [50], various quantum programming languages [1, 4, 19, 20, 22, 47, 48] have been proposed for more straightforward implementation of quantum algorithms. Very recently, significant efforts have been devoted to the research of quantum logic and quantum program analysis [14, 23, 43, 49, 55, 56, 63, 65] for these emerging quantum programs.

Comparison with Quantitative Robustness Reasoning. [26] develops semantics for erroneous quantum while programs and logic to prove robustness between an ideal program and a noisy one. [66] derives applied quantum Hoare logic by employing projection as predicates and reasons about the robustness of quantum programs, i.e., error bounds of outputs. These two works focus on single-program executions, while our work studies relational reasoning. In particular, the major differences are as follows. a). Different formula: In the logic formula of [26, 66], the predicate lives in the space of the principle program. The predicate of our logic lives in the joint space of the two programs. b). Different scope of applications: The proof systems developed by [26, 66] focus on studying the robustness of quantum programs, i.e., equivalence or closeness. Our choice of relational Hoare logic can reason about general relations beyond equivalence or closeness. We can reason relational properties between programs with different numbers of qubits. c). Different proof rules: The proof rules of [26, 66] discuss programs with the same syntax statement, while our one-side rules can track relational properties for different statements.

Comparison with Relational Quantum Hoare Logics. Our work is primarily inspired by the quantum relational Hoare logics recently proposed by [8, 33, 51]. In particular, [8] suggests that casting approximate reasoning into the general framework of relational quantum Hoare logic remains open. Generally, two quantum while programs do not share the same probabilities for taking different paths or outcomes during their execution. Under such circumstances, exact quantum couplings cannot be found, as they only exist for partial density operators with identical trace. This mathematical condition significantly restricts the flexibility of the exact quantum relational Hoare logic. Our work provides a promising solution to this open question. In particular, by introducing approximate quantum coupling, our logic system offers a more general scope of applications. Our logic, aqRHL, is a quantum counterpart to apRHL [9], even from a technical point of view: aqRHL employs projective predicates [12] over the joint systems of the programs, a natural quantum counterpart of binary relations, the predicates used in apRHL.

Future Work. There are several promising directions for future work. Firstly, we would like to extend our theory to the hybrid system, i.e., programs with quantum and classical variables. Hybrid quantum-classical systems allow for the exploitation of quantum advantages while leveraging the existing classical computing infrastructure. A unified language incorporating both quantum and classical effects may offer advantages in analyzing hybrid programs [52]. Secondly, we will investigate the potential applications of the newly developed approximate relational quantum Hoare logic. Particularly, we are interested in applying it to the construction and verification of quantum cryptographic proofs and ensuring the correctness of optimized quantum compilers specifically designed for NISQ (Noisy Intermediate-Scale Quantum) devices. Lastly, it is interesting to incorporate recently developed tools such as quantum abstract interpretation [62] and quantum separation logic [64] to design over-approximation techniques [58]. Another interesting technique is Context-Free-Language Ordered Binary Decision Diagrams [46], which may serve as a backend representation and manipulation technique in studying quantum Hoare logics.

9 Conclusion

We resolve the open question of [8] by designing an approximate relational Hoare logic for robustly reasoning the relational properties of two programs. We showcase the success of our methodology by formally verifying the well-known low-depth approximation of the quantum Fourier transform, and the correctness of the repeat-until-success algorithm and bit flip code.